M. N. Ivanov.

ON SMALL OSCILLATIONS

OF A MATERIAL SYSTEM

near an equilibrium position.

Tomsk.

Zinc-lithography of the Siberian T-va [[unclear: imprint abbreviation]], Printing House, corner of Dvoryanskaya St. and Yamsk. Lane, own house.

1916.

On Small Oscillations of a Material System near an Equilibrium Position

INTRODUCTION.

Oscillatory motions of a strictly or approximately harmonic type, owing to their close connection with many important questions of both theoretical and applied physics, quite early became the subject of special study. The investigation of the simplest cases of oscillations of mathematical and physical pendulums belongs to Galileo (1564—1642) and Huygens (1629—1695)—the founders of dynamics; in the analytical mechanics of Lagrange (1736—1813) we already see a developed theory of the oscillatory motions of a material system near a position of equilibrium. In its main features this theory has been preserved up to the present time; the works of later mathematicians (Cauchy, Sturm, Weierstrass, Somov) perfected separate parts of Lagrange’s theory and corrected an error he had made, without changing the method of investigation itself.

An exposition of Lagrange’s theory of small oscillations, with the corrections and additions of later times, may be found in very well-known English treatises: Treatise on Natural Philosophy by Lord Kelvin and Tait, Theory of Sound by Lord Rayleigh, and Dynamics of a System of Rigid Bodies by Routh, as well as in the work of Prof. Umov entitled “From Lectures on Mathematical Physics.” Written much earlier, the divisions of mechanical systems into holonomic and nonholonomic, introduced by Hertz, were not taken into account in the above-mentioned works, which have in view the oscillations only of holonomic systems. This is their first distinctive feature.

Another distinctive feature is the removal from the exposition of many details of an algebraic character, which makes comprehension difficult and compels the reader to make up the missing details from extraneous sources.

The aim of the present article is to give an exposition of the corrected theory of Lagrange, extending it to nonholonomic systems, using—

using, in doing so, the methods of algebra and analysis to such an extent as is indicated, on the one hand, by the rigor of the mathematical exposition, and, on the other, by its possible simplicity and clarity.

General formulation of the question.

The mechanical systems whose small oscillations we intend to study are assumed to consist of separate material particles with masses \(m_i\) (so-called material points) and coordinates \(x_i, y_i, z_i\), where the index \(i\) runs through the series of values from \(1\) to \(m\), equal to the number of all distinct particles of the system.

In full agreement with the above-mentioned authors, we shall assume that the forces acting on the system are conservative, i.e. depend only on the position of the points, but not on the velocities; as regards the character of the kinematic constraints between the various points of the system, we shall adhere to more general assumptions: alongside constraints expressible by finite equations between the coordinates of the form

\[ \varphi_i(x_1,y_1,z_1,x_2,y_2,z_2,\ldots x_m,y_m,z_m)=0, \]

we shall also allow the existence of differential constraints, expressed, as always, by equations linear with respect to the first derivatives of all, or only some, coordinates with respect to time.

In other words, we shall have in view systems both holonomic and nonholonomic, but with one essential restriction: that the time \(t\) does not enter the equations of constraint explicitly, but only through the coordinates of the various points of the system, or their first derivatives with respect to time, i.e. the components of the velocities of the separate points along the coordinate axes. The absence of the time \(t\) in the equations of constraint in explicit form excludes the occurrence, with the passage of time, of any changes either in the constraints themselves as a whole or, in particular, in that equilibrium configuration about which, by assumption, the small oscillations of the system are to take place.

Extension of Lagrange’s method to nonholonomic systems.

The possibility of extending the theory of infinitesimal oscillations to nonholonomic systems follows from the fact that, when only infinitesimal displacements are admitted, every differential constraint of the above-indicated form (the so-called scleronomic one) becomes an integrable constraint and, consequently, may be replaced by a finite equation between the coordinates.

To see this, let us temporarily leave aside the differential constraints of the non-holonomic system, which we shall suppose to consist of \(m\) discrete material particles, or points, connected by finite equations of the form:

\[ \varphi_i(x_1,y_1,z_1,\ldots x_m,y_m,z_m)=0 \quad (i=1,2,3,\ldots i), \tag{1} \]

and by differential equations of the form:

\[ \psi_k(x_1,y_1,\ldots x_i',y_i',z_i',\ldots) \equiv A_{k1}x_1' + B_{k1}y_1' + C_{k1}z_1' + \ldots + A_{km}x_m' \]

\[ {}+B_{km}y_m' + C_{km}z_m'=0 \quad (k=1,2,3,\ldots k), \tag{2} \]

and for the present we shall consider only its finite constraints \(\varphi_i=0\).

Then, according to the general rule applicable to every holonomic system, of the \(3m\) Cartesian coordinates of the separate material points of this system, connected by \(i\) finite equations (1) of constraints, only \(3m-i\) coordinates, chosen at our discretion, will be completely independent; the remaining \(i\) coordinates will be certain functions of the first \(3m-i\) coordinates.

Whatever may be the form of the functions expressing the dependence of \(i\) coordinates on the remaining coordinates of the system, these functions will always possess one evident property, namely: when substituted into the equations of constraint, they will reduce these equations identically to zero, i.e. for any system of particular values of the independent coordinates. It is clear that this property will also be preserved if we replace the system of \(3m-i\) mutually independent coordinates by some other system of likewise mutually independent coordinates \(q_1,q_2,q_3,\ldots q_{3m-i}\), subject to the sole condition that there exist a one-to-one correspondence between the first and the second coordinates.

Having established, in one way or another, a relation between the \(3m-i\) independent coordinates of the old system and the same number of new coordinates, which, in distinction from the former, we shall call independent parameters or generalized coordinates of Lagrange, we shall without difficulty express also the remaining \(i\) dependent coordinates through the same parameters \(q_i\), and in this way obtain the following system of equations connecting the old coordinates with the new:

\[ \begin{aligned} x_i&=\overline{x}_i(q_1,q_2,\ldots q_n)\\ y_i&=\overline{y}_i(q_1,q_2,\ldots q_n)\\ z_i&=\overline{z}_i(q_1,q_2,\ldots q_n) \end{aligned} \left\{ \quad i=1,2,3,\ldots m \right. \tag{3} \]

where by \(n\) is denoted, for brevity, the number \(3m-i\), representing the number of independent coordinates of the system, or the so-called number of degrees of freedom of the holonomic system. As was just said, in the choice of the new coordinates two requirements were taken into account: 1) that the finite equations of constraint, upon substitution into them of the new coordinates, should turn into identities, and 2) that every system of particular values of the coordinates \(q_i(i=1,2,3\ldots n)\) could completely determine the corresponding position of the system. To these requirements is added still a third, following from the continuity of the motion: that the coordinates \(x,y,z\) of the various points of the system be continuous functions of the coordinates \(q_i\), and that the \(q\) themselves be continuous functions of time.

Without impairing the generality of the investigation, one may, for the sake of simplifying the exposition, assume that the independent variables \(q_i\) are reckoned from the position of equilibrium,—in other words, one may assume that in the position of equilibrium all the parameters \(q_1,q_2,\ldots q_n\) are equal to zero.

Then, expanding each of the coordinates \(x_i,y_i,z_i\) of any point of the system near the position of equilibrium, we obtain for these coordinates the following expressions:

\[ \begin{aligned} x_i&=\xi_i+\left(\frac{\partial x_i}{\partial q_1}\right)_0\Delta q_1+ \left(\frac{\partial x_i}{\partial q_2}\right)_0\Delta q_2+\cdots =\xi_i+\sum_{k=1}^{k=n}\left(\frac{\partial x_i}{\partial q_k}\right)_0\Delta q_k+\alpha_2,\\ \tag{4} y_i&=\eta_i+\left(\frac{\partial y_i}{\partial q_1}\right)_0\Delta q_1+ \left(\frac{\partial y_i}{\partial q_2}\right)_0\Delta q_2+\cdots =\eta_i+\sum_{k=1}^{k=n}\left(\frac{\partial y_i}{\partial q_k}\right)_0\Delta q_k+\beta_2,\\ z_i&=\zeta_i+\left(\frac{\partial z_i}{\partial q_1}\right)_0\Delta q_1+ \left(\frac{\partial z_i}{\partial q_2}\right)_0\Delta q_2+\cdots =\zeta_i+\sum_{k=1}^{k=n}\left(\frac{\partial z_i}{\partial q_k}\right)_0\Delta q_k+\gamma_2. \end{aligned} \]

In these formulas, by \(\xi_i,\eta_i,\zeta_i\) are denoted the particular values of \(x_i,y_i,z_i\) when into them are substituted \(q_1=q_2=q_3=\cdots=q_n=0\), i.e. the Cartesian coordinates of the various points of the system in the position of equilibrium, and by \(\alpha_2,\beta_2,\gamma_2\)—the aggregate of terms of the 2nd and higher orders in each of the expansions.

In small oscillations the deviations of the system from the position of equilibrium remain small quantities, and therefore, in the first approximation, it is possible in the expansion of the coordinates to confine oneself to terms of the first order only. For convenience we shall write in the expansions simply \(q_k\) instead of \(\Delta q_k\), assuming herewith that the quantities \(q_k\) remain so small that their squares may be

neglect. In such a case, equations (4) can be given a more convenient form:

\[ \left. \begin{aligned} x_i-\xi_i&=\sum_{k=1}^{k=n}\left(\frac{\partial x_i}{\partial q_k}\right)_0 q_k,\\ y_i-\eta_i&=\sum_{k=1}^{k=n}\left(\frac{\partial y_i}{\partial q_k}\right)_0 q_k,\\ z_i-\zeta_i&=\sum_{k=1}^{k=n}\left(\frac{\partial z_i}{\partial q_k}\right)_0 q_k \end{aligned} \right\} \quad i=1,2,3\ldots m. \tag{5} \]

Turning now to the differential constraints which were temporarily left out of consideration, and transforming the linear equations (1) to the new variables \(q_i\) with the aid of equations (5) and their derivatives, we obtain equations of the same linear form with respect to the derivatives \(q'_1,q'_2,\ldots,q'_n\), with coefficients depending on the parameters \(q_i\) and on the coefficients of the transformation. Expanding each of the coefficients in powers of the small quantities \(q_i\), we obtain the transformed equations of the differential constraints in the form:

\[ \sum_{i=1}^{i=n} C_i^k q'_i+\sum_{ij} C_{ij}^k q'_i q_j+\text{terms of the 2nd and higher orders}=0, \tag{6} \]

where

\[ k=1,2,3\ldots k,\quad \text{and}\quad i=j=1,2,3\ldots n. \]

In these equations, according to Lagrange’s theory, subsequently more precisely established by L. Dirichlet, when small oscillations about a position of stable equilibrium are under consideration, not only the coordinates \(q_i\), but also their time derivatives, or velocities \(q'_i\), remain small quantities; therefore, with accuracy up to quantities of the second order, we have the right to write the equations of the differential constraints in the simpler form:

\[ \sum_{i=1}^{i=n} C_i^k q'_i=0,\quad (k=1,2,3\ldots k). \tag{7} \]

All the coefficients of equations (7) are constant, and therefore the left-hand sides of these equations admit termwise integration, which transfers the differential constraints to the class of finite constraints.

The arbitrary constants introduced by integrating the equations of constraint must all be set equal to zero, since in the position of equilibrium all the parameters \(q_i\), by assumption, are equal to zero and satisfy the equations of constraint.

Thus it is clear that every differential constraint in the present case (i.e., for small oscillations) acts on the system in the same way as a finite constraint; that is, it lowers by one the number of independent coordinates, or, what is the same thing, reduces by one the number of degrees of freedom of the system; so that the result of introducing \(k\) mutually independent differential constraints is the loss by the system of another \(k\) degrees of freedom.

Formulation of the differential equations of motion of the system.

In view of all that has been said above, we shall assume the position of any system, holonomic or nonholonomic, to be given by equations of the form:

\[ \begin{gathered} x_i=\xi_i+\sum_k \alpha_{ik}q_k,\\ \\ y_i=\eta_i+\sum_k \beta_{ik}q_k,\\ \\ z_i=\zeta_i+\sum_k \gamma_{ik}q_k. \end{gathered} \tag{8} \]

In these equations \(\xi_i,\eta_i,\zeta_i\) are the values of the coordinates of the separate points of the system in the position of stable equilibrium, while the quantities \(q_i\) are independent Lagrange parameters whose number is equal to the number of degrees of freedom of the moving system; all \(q_k\), by the conditional choice of the origin, are equal to zero in the position of equilibrium and differ from zero, but remain small quantities together with their time derivatives, during the whole time of motion; the coefficients \(\alpha_{ik},\beta_{ik},\gamma_{ik}\) are constants, namely the partial derivatives

\[ \left(\frac{\partial x_i}{\partial q_k}\right)_0,\quad \left(\frac{\partial y_i}{\partial q_k}\right)_0,\quad \left(\frac{\partial z_i}{\partial q_k}\right)_0. \]

To formulate the equations of motion in Lagrange’s form, it is necessary to have two expressions: the expression for the kinetic energy and the potential function of the system.

Assuming these expressions, if they are given directly as functions of the coordinates \(x_i,y_i,z_i\), transformed to the new variables \(q_k\) and their derivatives, we shall give them the following form:

\[ 2T=\sum_i m_i\left(x_i'^2+y_i'^2+z_i'^2\right)=\sum_{ik} a_{ik}q_i'q_k', \]

\[ 2V=2V\left\{\xi_i+\sum_k \alpha_{ik}q_k,\ \eta_i+\sum_k \beta_{ik}q_k,\ \zeta_i+\sum_k \gamma_{ik}q_k,\ldots\right\}. \]

Of these two formulas, the first is obtained by means of elementary algebraic operations and represents a homogeneous quadratic function of the rates of change of the unknown parameters \(q_i'\), with coefficients \(a_{ik}\), which are composed in a very simple way from the coefficients \(\alpha_{ik}, \beta_{ik}, \gamma_{ik}\). As for the second formula for the potential function \(2V\), expanding it in powers of the various \(q_k\), we first obtain the formula:

\[ \begin{aligned} 2V={}&2V_0+2\left\{\left(\frac{\partial V}{\partial q_1}\right)_0 q_1+ \left(\frac{\partial V}{\partial q_2}\right)_0 q_2+\cdots\right\} \\ &+\left\{\left(\frac{\partial^2 V}{\partial q_1^2}\right)_0 q_1^2 +2\left(\frac{\partial^2 V}{\partial q_1\partial q_2}\right)_0 q_1q_2 +2\left(\frac{\partial^2 V}{\partial q_1\partial q_3}\right)_0 q_1q_3+\cdots\right.\\ &\left.\qquad+\left(\frac{\partial^2 V}{\partial q_2^2}\right)_0 q_2^2 +\cdots+ \left(\frac{\partial^2 V}{\partial q_k^2}\right)_0 q_k^2\right\} +\alpha_3, \end{aligned} \]

in which \(\alpha_3\) denotes the aggregate of the terms of the expansion of the 3rd and higher orders. The partial derivatives of \(V\) with respect to the various coordinates \(q_i\) in the equilibrium position, i.e. all \(\left(\dfrac{\partial V}{\partial q_i}\right)_0\), must vanish, which brings \(2V\) to the form:

\[ 2V=2V_0+\sum_{ik} b_{ik}q_iq_k, \tag{9} \]

if one retains only the terms of the 2nd order of smallness and writes, in abbreviated form, the symbol \(b_{ik}\) instead of \(\left(\dfrac{\partial^2 V}{\partial q_i\,\partial q_k}\right)_0\).

The expressions obtained for the living force and the potential energy in some position of the system close to the position of stable equilibrium make it possible to write the differential equations of motion of this system.

The general form of these equations

\[ \frac{d}{dt}\left(\frac{\partial T}{\partial q_i'}\right)-\frac{\partial T}{\partial q_i} = -\frac{\partial V}{\partial q_i} \quad (i=1,2,\ldots,k) \]

in the present case is reduced to the simpler one:

\[ \frac{d}{dt}\left(\frac{\partial T}{\partial q_i'}\right)+\frac{\partial V}{\partial q_i}=0 \quad (i=1,2,3,\ldots,k), \tag{10} \]

because the expression for the kinetic energy contains no parameters \(q_i\), and therefore partial derivatives of the form \(\left(\dfrac{\partial T'}{\partial q_i}\right)\) are equal to zero for every index \(i\).

Writing out in detail the expression

\[ \frac{\partial T}{\partial q_i'}=a_{i1}q_1' + a_{i2}q_2' + a_{i3}q_3' + \cdots + a_{ik}q_k', \]

we immediately observe that

\[ \frac{\partial T}{\partial q_i'}=\frac{d}{dt}\left(a_{i1}q_1+a_{i2}q_2+a_{i3}q_3+\cdots+a_{ik}q_i\right), \]

where the polynomial

\[ a_{i1}q_1+a_{i2}q_2+a_{i3}q_3+\cdots+a_{ik}q_k \]

is the partial derivative with respect to \(q_i\) of a certain homogeneous quadratic function

\[ Q=\frac{1}{2}\sum_{ik} a_{ik}q_iq_k, \]

which is obtained from the expression for the kinetic energy \(T\) by simply dropping the derivative signs from the parameters \(q_1,q_2,q_3\ldots q_k\).

For the sake of brevity, we shall agree to denote the partial derivatives of the homogeneous quadratic functions \(Q\) and \(V-V_0\) with respect to the argument \(q_i\), respectively, by \(Q_i\) and \(V_i\), so that

\[ Q_i=\frac{\partial Q}{\partial q_i}=a_{i1}q_1+a_{i2}q_2+a_{i3}q_3+\cdots+a_{ik}q_k, \tag{11} \]

\[ V_i=\frac{\partial V}{\partial q_i}=b_{i1}q_1+b_{i2}q_2+b_{i3}q_3+\cdots+b_{ik}q_k. \tag{12} \]

Then equations (10) may be given the more symmetric form:

\[ \frac{d^2 Q_i}{dt^2}+V_i=0 \quad (i=1,2,3,\ldots k). \tag{13} \]

System (13) is a system of linear homogeneous differential equations of the second order with constant coefficients; its integration is usually carried out by substituting into the differential equations of the system trial solutions of the form \(q_i=A_i e^{st}\), and then finding the values of the multiplier \(s\), which determine the so-called periods of oscillation of the system—quantities closely

connected with the very nature of the system and unable to change depending on one or another choice of the system of coordinates.

But these same equations can be integrated by another method, not adopted by the authors who have treated the question of small oscillations, but which, in my opinion, has certain advantages over the usual method, consisting in greater naturalness and accessibility for understanding both the general plan and the individual details of the solution.

Transformation of Lagrange’s Equations of Motion into Equations of Harmonic Type.

Lagrange’s equations (13) only outwardly resemble the equations of simple harmonic oscillations, since the functions \(Q_i\) and \(V_i\) differ from one another; but, using the linearity of these functions with respect to the variables \(q_i\), we can, following D’Alembert’s method, seek such a linear combination of equations (13) as would indeed constitute an equation of harmonic type.

For this purpose we multiply the various equations of the system (13) successively by undetermined multipliers \(\alpha_1,\alpha_2,\alpha_3,\ldots,\alpha_k\) and add them term by term, bringing the undetermined, but time-independent, multipliers \(\alpha_i\) under the sign of the second derivative. The result of these operations will be an equation of the form:

\[ \frac{d^2}{dt^2}\{\alpha_1 Q_1+\alpha_2 Q_2+\alpha_3 Q_3+\cdots+\alpha_k Q_k\} +\{\alpha_1 V_1+\alpha_2 V_2+\cdots+\alpha_k V_k\}=0. \tag{14} \]

Using the indeterminacy of the multipliers \(\alpha_i\), we can determine them so that the ratio

\[ \frac{\alpha_1 V_1+\alpha_2 V_2+\alpha_3 V_3+\cdots+\alpha_k V_k} {\alpha_1 Q_1+\alpha_2 Q_2+\alpha_3 Q_3+\cdots+\alpha_k Q_k}=s \]

does not depend on the variables \(q_i\) and the time \(t\), in other words, so that the equality

\[ \alpha_1 V_1+\alpha_2 V_2+\cdots+\alpha_k V_k =s(\alpha_1 Q_1+\alpha_2 Q_2+\alpha_k Q_k) \]

be fulfilled for arbitrary values of \(q_i\) and \(t\).

This requirement is satisfied when the coefficients of the various \(q_i\) in the right- and left-hand sides of the equality are proportional, i.e. it presupposes the existence of the following series of equalities:

\[ \left. \begin{aligned} &b_{11}\alpha_1+b_{21}\alpha_2+b_{31}\alpha_3+\ldots+b_{k1}\alpha_k =s(a_{11}\alpha_1+a_{21}\alpha_2+a_{31}\alpha_3+\ldots a_{k1}\alpha_k),\\ &b_{12}\alpha_1+b_{22}\alpha_2+b_{32}\alpha_3+\ldots+b_{k2}\alpha_k =s(a_{12}\alpha_1+a_{22}\alpha_2+a_{32}\alpha_3+\ldots a_{k2}\alpha_k),\\ &b_{13}\alpha_1+b_{23}\alpha_2+b_{33}\alpha_3+\ldots+b_{k3}\alpha_k =s(a_{13}\alpha_1+a_{23}\alpha_2+a_{33}\alpha_3+\ldots a_{k3}\alpha_k),\\ &\cdots\\ &\cdots\\ &b_{1k}\alpha_1+b_{2k}\alpha_2+b_{3k}\alpha_3+\ldots+b_{kk}\alpha_k =s(a_{1k}\alpha_1+a_{2k}\alpha_2+a_{3k}\alpha_3+\ldots a_{kk}\alpha_k). \end{aligned} \right\} \tag{15} \]

Transferring all terms to the right-hand side and arranging them according to the arbitrary multipliers \(\alpha_i\), we shall have a series of equations satisfying the condition as a function of \(s\):

\[ \left. \begin{aligned} &(a_{11}s-b_{11})\alpha_1+(a_{21}s-b_{21})\alpha_2+(a_{31}s-b_{31})\alpha_3+\ldots+(a_{k1}s-b_{k1})\alpha_k=0,\\ &(a_{12}s-b_{12})\alpha_1+(a_{22}s-b_{22})\alpha_2+(a_{32}s-b_{32})\alpha_3+\ldots+(a_{k2}s-b_{k2})\alpha_k=0,\\ &(a_{13}s-b_{13})\alpha_1+(a_{23}s-b_{23})\alpha_2+(a_{33}s-b_{33})\alpha_3+\ldots+(a_{k3}s-b_{k3})\alpha_k=0,\\ &\cdots\\ &\cdots\\ &(a_{1k}s-b_{1k})\alpha_1+(a_{2k}s-b_{2k})\alpha_2+(a_{3k}s-b_{3k})\alpha_3+\ldots+(a_{kk}s-b_{kk})\alpha_k=0. \end{aligned} \right\} \tag{16} \]

The condition for the compatibility of these equations is that the determinant of this system be equal to zero:

\[ \Delta(s)=\|a_{11}s-b_{11},\ a_{22}s-b_{22},\ a_{33}s-b_{33},\ldots,\ a_{kk}s-b_{kk}\|. \tag{17} \]

For brevity we shall call both the determinant \(\Delta(s)\) and the system of multipliers \(\alpha_i\) harmonizing, since by means of them equation (14) is brought to harmonic form.

Leaving aside the difficulties of solving the equation \(\Delta(s)=0\), and supposing that some root of this equation has been found, \(s=s_h\), we shall transform the system of conditionally compatible equations (16) into a system of actually compatible equations by replacing, in equations (16), the indeterminate multiplier \(s\) by \(s_h\).

From the linear homogeneous equations thus obtained with respect to the multipliers \(\alpha_i\), one can determine a series of quantities proportional to these multipliers \(\alpha_i\), which we shall denote by the same index \(h\) that stands by the root \(s_h\).

In this way the following series of ratios is obtained:

\[ \frac{\alpha_{1h}}{M_{j1}(s_h)} = \frac{\alpha_{2h}}{M_{j2}(s_h)} = \frac{\alpha_{3h}}{M_{j3}(s_h)} = \ldots = \frac{\alpha_{kh}}{M_{jk}(s_h)} = C_{jh}, \]

where \(M_{jn}\) is the minor of the determinant \(\Delta(s)\) corresponding to the element situated at the intersection of the \(j\)-th row and the \(n\)-th column, and \(C_{jh}\) is a mul-

factor of proportionality; the indices of the constant \(C\) denote: the first—the row, the second—the root \(s_h\).

It is clear that, instead of the minors corresponding to the elements of the \(j\)-th row of the determinant, one could take the minors of the elements of any other row of the same determinant, since between the minors of the determinant \(\Delta(s)\), which vanishes identically when one of the roots \(s_h\) is substituted for \(s\), there exists the following relation:

\[ \frac{M_{i1}}{M_{j1}} = \frac{M_{i2}}{M_{j2}} = \frac{M_{i3}}{M_{j3}} = \cdots = \frac{M_{ik}}{M_{jk}}, \]

expressing the proportionality between the minors of elements belonging to one and the same column (row) in two different rows (columns).

With regard to all minors in general of the determinant \(\Delta(s)\), one may note that they are integral rational functions of \(s\); therefore all minors, and consequently all factors \(\alpha_i\) proportional to these minors, will be real quantities when real roots of the equation \(\Delta(s)=0\) are substituted for \(s\), and imaginary quantities when imaginary roots are substituted.

The various methods by which equations of harmonic type can be formed from Lagrange’s equations (13) can depend only on the real roots of the equation \(\Delta(s)=0\), and therefore the question of the nature of the roots of the determinant \(\Delta(s)\) occupies a prominent place in the theory of small oscillations of a system.

The absence of general criteria allowing one to judge the nature of the roots of equations with literal coefficients has forced one to seek a solution of this problem by indirect means, as will be seen from the following exposition.

The reality of the roots of the equation \(\Delta(s)=0\).

The method of proof of the reality of the roots of the determinant \(\Delta(s)\) that I present is based on investigations of Cauchy, Jacobi, Weierstrass, Sylvester, Somov, and others, concerning the transformation of quadratic forms and the equation of secular inequalities. In justification of the deviations from the accepted order and form of exposition, I must say that my aim was at the same time both to prove the reality of the roots and to indicate, in general outline, the path by which these roots, if desired, can actually be found.

The proof of the principal proposition is composed of three equally important parts: 1) the proof of the invariance of the roots of the determinant \(\Delta(s)\) after simplifying it by means of a linear transformation, 2) the proof of the possibility of simplifying the external form of the determinant, and 3) the proof of the reality of the roots of the determinant of the given simplified form.

1) Invariance of the roots of the harmonizing determinant.

For the convenience of the further exposition, let us agree to denote the results of substituting, in the quadratic functions \(Q\) and \(V - V_0\), in place of the arguments \(q_1, q_2, \ldots, q_k\), the factors \(\alpha_1, \alpha_2, \alpha_3, \ldots, \alpha_k\), respectively, by \(\varphi(\alpha_1, \alpha_2, \ldots, \alpha_k)\) and \(\psi(\alpha_1, \alpha_2, \ldots, \alpha_k)\), or still more briefly by \(\varphi\) and \(\psi\), without indicating the arguments; and the partial derivatives of the first and second orders of \(\varphi\) and \(\psi\), considered as functions of the independent variables \(\alpha_1, \alpha_2, \ldots, \alpha_k\), shall be denoted by the symbols:

\[ \varphi_1, \psi_1, \varphi_2, \psi_2, \ldots, \varphi_{11}, \varphi_{12}, \ldots \ \text{etc.}, \quad \text{where } \varphi_k = \frac{\partial \varphi}{\partial \alpha_k}, \quad \text{and } \varphi_{ik} = \frac{\partial^2 \varphi}{\partial \alpha_i \partial \alpha_k}. \]

With the aid of these notations, under the condition that \(s\) does not depend on the variables \(\alpha_1, \alpha_2, \ldots, \alpha_k\), the harmonizing determinant \(\Delta(s)\) may be represented in the following form:

\[ \begin{aligned} \Delta(s) &= \left\| \frac{\partial^2(s\varphi-\psi)}{\partial \alpha_1^2}, \frac{\partial^2(s\varphi-\psi)}{\partial \alpha_2^2}, \frac{\partial^2(s\varphi-\psi)}{\partial \alpha_3^2}, \ldots, \frac{\partial^2(s\varphi-\psi)}{\partial \alpha_k^2} \right\| \\ &= \left\| \Phi_{11}, \Phi_{22}, \Phi_{33}, \ldots, \Phi_{kk} \right\|, \end{aligned} \tag{18} \]

where, for brevity, it is put that \(s\varphi - \psi = \Phi\), and the symbol \(\Phi_{ik}\) denotes the second derivative

\[ \frac{\partial^2 \Phi}{\partial \alpha_i \partial \alpha_k}. \]

The determinant \(\left\| \Phi_{11}, \Phi_{22}, \Phi_{33}, \ldots, \Phi_{kk} \right\|\) may also be given another form:

\[ \mathrm{H}(\Phi)_{\alpha_1 \alpha_2 \ldots \alpha_k} = \left\| \frac{\partial \Phi_1}{\partial \alpha_1}, \frac{\partial \Phi_2}{\partial \alpha_2}, \frac{\partial \Phi_3}{\partial \alpha_3}, \ldots, \frac{\partial \Phi_k}{\partial \alpha_k} \right\|, \]

that is, the so-called Hessian of the function \(\Phi\) with respect to its arguments \(\alpha_1, \alpha_2, \alpha_3, \ldots, \alpha_k\).

From the manner of forming the Hessian it follows, as a consequence, that every transformation performed on its functions must take the form of a Hessian. Without touching upon the general question of how an arbitrary transformation of its functions is reflected in the composition of the Hessian, we shall consider here the particular question of the change in the form of the Hessian as a function of a linear transformation performed on its functions. In order—

To formulate our problem more precisely, suppose that the function \(\Phi(\alpha_1,\alpha_2,\alpha_3\ldots \alpha_k)\) is obtained from another homogeneous and quadratic function \(\Phi'(\beta_1,\beta_2,\beta_3\ldots \beta_k)\), whose arguments \(\beta_i\) are connected with the arguments of the first function \(\alpha_i\) by homogeneous linear equations with real coefficients \(l_1,l_2\ldots r_1,r_2,\ldots r_k\) of the following form:

\[ \left. \begin{aligned} \beta_1 &= l_1\alpha_1+l_2\alpha_2+l_3\alpha_3+\cdots+l_k\alpha_k,\\ \beta_2 &= m_1\alpha_1+m_2\alpha_2+m_3\alpha_3+\cdots+m_k\alpha_k,\\ \beta_3 &= n_1\alpha_1+n_2\alpha_2+n_3\alpha_3+\cdots+n_k\alpha_k,\\ &\cdots\\ &\cdots\\ \beta_k &= r_1\alpha_1+r_2\alpha_2+r_3\alpha_3+\cdots+r_k\alpha_k, \end{aligned} \right\} \tag{19} \]

and it is required to establish the relation between the Hessians

\[ \mathrm{H}(\Phi)_{\alpha_1\alpha_2\ldots \alpha_k} \qquad \text{and} \qquad \mathrm{H}(\Phi')_{\beta_1\beta_2\ldots \beta_k}. \]

To solve the problem posed, we shall compute the individual elements of the Hessian:

\[ \mathrm{H}(\Phi)_{\alpha_1\alpha_2\ldots \alpha_k} = \left\| \frac{\partial \Phi_1}{\partial \alpha_1}, \frac{\partial \Phi_2}{\partial \alpha_2}, \frac{\partial \Phi_3}{\partial \alpha_3}, \ldots \frac{\partial \Phi_k}{\partial \alpha_k} \right\|, \]

regarding the function \(\Phi\) as a composite function equal to \(\Phi'(\beta_1,\beta_2\ldots \beta_k)\), in which the arguments \(\beta\) are replaced by their values by means of equations (19). Using the rules for differentiating composite functions, we first find the expressions \(\Phi_1,\Phi_2,\ldots \Phi_k\), which have the following form:

\[ \Phi_1 = \frac{\partial \Phi}{\partial \alpha_1} = \frac{\partial \Phi'}{\partial \beta_1}\frac{\partial \beta_1}{\partial \alpha_1} + \frac{\partial \Phi'}{\partial \beta_2}\cdot \frac{\partial \beta_2}{\partial \alpha_1} +\cdots+ \frac{\partial \Phi'}{\partial \beta_k}\cdot \frac{\partial \beta_k}{\partial \alpha_1} = \Phi'_1 l_1+\Phi'_2 m_1\ldots+\Phi'_k r_1 \]

\[ \Phi_2 = \frac{\partial \Phi}{\partial \alpha_2} = \frac{\partial \Phi'}{\partial \beta_1}\frac{\partial \beta_1}{\partial \alpha_2} + \frac{\partial \Phi'}{\partial \beta_2}\cdot \frac{\partial \beta_2}{\partial \alpha_2} +\cdots+ \frac{\partial \Phi'}{\partial \beta_k}\cdot \frac{\partial \beta_k}{\partial \alpha_2} = \Phi'_1 l_2+\Phi'_2 m_2\ldots+\Phi'_k r_2. \]

\[ \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots \]

\[ \Phi_k = \frac{\partial \Phi}{\partial \alpha_k} = \frac{\partial \Phi'}{\partial \beta_1}\frac{\partial \beta_1}{\partial \alpha_k} + \frac{\partial \Phi'}{\partial \beta_2}\cdot \frac{\partial \beta_2}{\partial \alpha_k} +\cdots+ \frac{\partial \Phi'}{\partial \beta_k}\cdot \frac{\partial \beta_k}{\partial \alpha_k} = \Phi'_1 l_k+\Phi'_2 m_k\ldots+\Phi'_k r_k. \]

By the same rule, any one of the derivatives of the second order will have the form:

\[ \Phi_{ij} = \frac{\partial \Phi_i}{\partial \alpha_j} = \frac{\partial \Phi_i}{\partial \beta_1}\frac{\partial \beta_1}{\partial \alpha_j} + \frac{\partial \Phi_i}{\partial \beta_2}\cdot \frac{\partial \beta_2}{\partial \alpha_j} +\cdots+ \frac{\partial \Phi_i}{\partial \beta_k}\cdot \frac{\partial \beta_k}{\partial \alpha_j}, \]

\[ = \frac{\partial \Phi_i}{\partial \beta_1}\cdot l_j + \frac{\partial \Phi_i}{\partial \beta_2}\cdot m_j +\cdots+ \frac{\partial \Phi_i}{\partial \beta_k}\cdot r_j. \]

where the multipliers

\[ \frac{\partial \Phi_i}{\partial \beta_1},\quad \frac{\partial \Phi_i}{\partial \beta_2},\ldots, \frac{\partial \Phi_i}{\partial \beta_k} \]

are sums of terms of the following form:

\[ \tag{20} \left\{ \begin{aligned} \frac{\partial \Phi_i}{\partial \beta_1} &= \frac{\partial}{\partial \beta_1} \left(\Phi_1' l_i+\Phi_2' m_i+\cdots+\Phi_k' r_i\right) = \Phi_{11}' l_i+\Phi_{21}' m_i+\Phi_{31}' n_i+\cdots+\Phi_{k1}' r_i,\\ \frac{\partial \Phi_i}{\partial \beta_2} &= \frac{\partial}{\partial \beta_2} \left(\Phi_1' l_i+\Phi_2' m_i+\cdots+\Phi_k' r_i\right) = \Phi_{12}' l_i+\Phi_{22}' m_i+\Phi_{32}' n_i+\cdots+\Phi_{k2}' r_i,\\ &\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\\ \frac{\partial \Phi_i}{\partial \beta_k} &= \frac{\partial}{\partial \beta_k} \left(\Phi_1' l_i+\Phi_2' m_i+\cdots+\Phi_k' r_i\right) = \Phi_{1k}' l_i+\Phi_{2k}' m_i+\Phi_{3k}' n_i+\cdots+\Phi_{kk}' r_i . \end{aligned} \right. \]

Multiplying, in order to form \(\Phi_{ij}\), equations (20): the first by \(l_j\), the second by \(m_j\), …, the \(k\)-th by \(r_j\), and adding the results in the vertical direction, taking outside the brackets the common multipliers \(l_i,m_i,n_i,\ldots,r_i\), we obtain for \(\Phi_{ij}\) the following formula:

\[ \Phi_{ji} = \frac{\partial^2 \Phi'}{\partial a_i\,\partial a_j} = l_i\left(\Phi_{11}' l_j+\Phi_{12}' m_j+\cdots+\Phi_{1k}' r_j\right) + m_i\left(\Phi_{21}' l_j+\Phi_{22}' m_j+\cdots+\Phi_{2k}' r_j\right) \]

\[ \tag{21} + n_i\left(\Phi_{31}' l_j+\Phi_{32}' m_j+\cdots+\Phi_{3k}' r_j\right) +\cdots+ r_i\left(\Phi_{k1}' l_j+\Phi_{k2}' m_j+\cdots+\Phi_{kk}' r_j\right). \]

To represent more clearly the law of formation of the term \(\Phi_{ij}\), standing at the intersection of the \(i\)-th horizontal row with the \(j\)-th vertical column in the determinant \(H(\Phi)_{a_1 a_2\ldots a_k}\), consider the determinant

\[ P = \left\| \Phi_{11}', \Phi_{22}', \ldots, \Phi_{kk}' \right\| \cdot \left\| l_1, m_2, n_3 \ldots r_k \right\| = \left\| p_{11}, p_{22}, p_{33}\ldots p_{kk} \right\|, \]

whose elements can be obtained by multiplying the elements of the horizontal rows of one determinant by the elements of the horizontal rows of the other determinant.

Comparison of the multipliers enclosed in brackets in formula (21) with the elements of the \(j\)-th column of the determinant \(P\) permits us to write for \(\Phi_{ij}\) the following expression:

\[ \Phi_{ij} = l_i p_{1j}+m_i p_{2j}+n_i p_{3j}+\cdots+r_i p_{kj}, \]

which is formed by multiplying the elements of the \(i\)-th horizontal row of the determinant

\[ D=\|l_1,m_2,n_3\ldots r_k\|, \]

called the modulus of the linear transformation, into the elements of the \(j\)-th vertical of the determinant \(P\).

Since, by such a law, the elements of the determinant equal to the product of the determinants \(D\) and \(P\) are formed, we arrive at the conclusion that the determinant formed from the elements \(\Phi_{ij}\) is the product \(D\cdot P\), or, what is the same thing,

\[ \begin{aligned} \mathrm{H}(\Phi)_{\alpha_1\alpha_2\ldots\alpha_k} &=\|\Phi_{11},\Phi_{22},\Phi_{33}\ldots\Phi_{kk}\| =D\cdot P =D^2\|\Phi'_{11},\Phi'_{22},\Phi'_{33},\ldots\Phi'_{kk}\| \\ &=D^2\mathrm{H}(\Phi')_{\beta_1\beta_2\beta_3\ldots\beta_k}. \end{aligned} \tag{22} \]

The formula obtained shows that the Hessians of two homogeneous quadratic functions, obtained one from the other by means of a linear transformation of the arguments, differ from one another by a factor equal to the square of the modulus of the transformation.

Observing that the quadratic functions—the original and the transformed one by means of a linear substitution—are linear functions of \(s\), and that their Hessians \(\mathrm{H}(\Phi)\) and \(\mathrm{H}(\Phi')\) are the discriminants of the functions \(\Phi\) and \(\Phi'\), having the form \(\Delta(s)\) and \(\Delta'(s)\), we may write, instead of (22), the following equality:

\[ \Delta(s)=D^2\cdot \Delta'(s), \tag{23} \]

from which it is seen that, when \(D\) is not equal to zero, the discriminants \(\Delta(s)\) and \(\Delta'(s)\), being integral polynomials of the same degree with respect to \(s\), i.e. the roots of the equations \(\Delta(s)=0\) and \(\Delta'(s)=0\), are one and the same. Thus the invariance of the roots of the harmonizing determinant after a linear transformation is fully proved, and we may proceed to the question of simplifying the external form of the discriminant.

Simplification of the discriminant by means of elementary transformations performed on the function \(\Phi\).

Without resorting to the solution of equations, one may considerably simplify the external form of the discriminant by reducing the coefficient of \(s\) in the function \(\Phi\), i.e. the quadratic function \(\varphi\), to the form of a sum of \(k\) squares. For such a reduction we shall use the elementary device of grouping the terms of the quadratic form \(2\varphi\), which we shall write in the following manner:

\[ 2\varphi =a_{11}\left\{\alpha_1^2+2\alpha_1\left(\frac{a_{12}}{a_{11}}\alpha_2+2\frac{a_{13}}{a_{11}}\alpha_3+\ldots+\frac{a_{1k}}{a_{11}}\alpha_k\right)\right\} +a_{22}\alpha_2^2+2a_{23}\alpha_2\alpha_3\ldots \tag{24} \]

Completing the terms in parentheses to a perfect square, we shall give the form \(2\varphi\) the following form:

\[ 2\varphi = a_{11}\left(\alpha_1+\frac{a_{12}}{a_{11}}\alpha_2+\frac{a_{13}}{a_{11}}\alpha_3+\cdots+\frac{a_{1\kappa}}{a_{11}}\alpha_\kappa\right)^2 +a_{22}\alpha_2^2+\cdots+a_{\kappa\kappa}\alpha_\kappa^2 \]
\[ -\frac{1}{a_{11}}(a_{12}\alpha_2+a_{13}\alpha_3+\cdots)^2 \]
\[ \tag{25} = a_{11}(\alpha_1+a_2\alpha_2+a_3\alpha_3+\cdots+a_\kappa\alpha_\kappa)^2 +2\chi(\alpha_2,\alpha_3,\ldots,\alpha_\kappa), \]

where

\[ a_2=\frac{a_{12}}{a_{11}},\qquad a_3=\frac{a_{13}}{a_{11}},\ldots,\qquad a_\kappa=\frac{a_{1\kappa}}{a_{11}}, \]

and \(2\chi\) denotes a homogeneous quadratic function of the arguments \(\alpha_2,\alpha_3\ldots\alpha_\kappa\), the coefficients of this function being computed directly by means only of the elementary operations of addition and multiplication. Applying the same method of grouping terms to the form \(2\chi\), we shall represent it in the form:

\[ \tag{26} 2\chi = p_2(\alpha_2+b_3\alpha_3+b_4\alpha_4+\cdots+b_\kappa\alpha_\kappa)^2 +2\omega(\alpha_3,\alpha_4,\ldots,\alpha_\kappa), \]

where \(b_3,b_4,\ldots b_\kappa\) are coefficients analogous to the coefficients \(a_2,a_3,\ldots a_\kappa\), and \(2\omega\) is a new quadratic form in the arguments \(\alpha_3,\alpha_4,\ldots\alpha_\kappa\). Continuing the transformation in the same way, we shall finally give the quadratic form \(2\varphi\) the following form:

\[ \tag{27} 2\varphi' = a_{11}\beta_1^2+p_2\beta_2^2+p_3\beta_3^2+\cdots+p_\kappa\beta_\kappa^2, \]

where, for brevity, by \(\beta_1,\beta_2,\beta_3\ldots\beta_\kappa\) are denoted the following expressions:

\[ \tag{28} \left\{ \begin{aligned} \beta_1&=\alpha_1+a_2\alpha_2+a_3\alpha_3+\cdots+a_\kappa\alpha_\kappa,\\ \beta_2&=*+\alpha_2+b_3\alpha_3+\cdots+b_\kappa\alpha_\kappa,\\ \beta_3&=*+*+\alpha_3+c_4\alpha_4+\cdots+c_\kappa\alpha_\kappa,\\ &\cdots\cdots\cdots\cdots\cdots\cdots\cdots\\ &\cdots\cdots\cdots\cdots\cdots\cdots\cdots\\ \beta_\kappa&=*+*+*+\cdots+\alpha_\kappa. \end{aligned} \right. \]

Since, for the computation of the coefficients in the substitutions (28), only multiplication and addition are used, they can only be real quantities.

The coefficients \(p_2,p_3,\ldots p_\kappa\), in view of their special role, may be given a form convenient for computation by making use of the above-mentioned property of the Hessians of quadratic forms.

Since the form \(2\varphi\) is obtained from \(2\varphi'\) by the linear substitutions (28), it follows that
\[ \mathrm{H}(\varphi)_{\alpha_1 \alpha_2 \ldots \alpha_\kappa} = \mathrm{H}(\varphi')_{\beta_1 \beta_2 \ldots \beta_\kappa} D_1^2, \]
where
\[ D_1= \begin{vmatrix} 1, & a_2, & a_3, & \ldots & a_\kappa\\ 0, & 1, & b_3, & b_4, & \ldots & b_\kappa\\ 0, & 0, & 1, & c_4, & \ldots & c_\kappa\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ 0, & 0, & 0, & 0, & \ldots & 1 \end{vmatrix} =1. \]

Consequently
\[ \mathrm{H}(\varphi)_{\alpha_1 \alpha_2 \ldots \alpha_\kappa} = \mathrm{H}(\varphi')_{\beta_1 \beta_2 \ldots \beta_\kappa} = \begin{vmatrix} a_{11} & 0 & 0 & 0 & \ldots & 0\\ 0 & p_2 & 0 & 0 & \ldots & 0\\ 0 & 0 & p_3 & 0 & \ldots & 0\\ \cdot & \cdot & \cdot & \cdot & \cdot & 0\\ 0 & 0 & 0 & 0 & \ldots & p_\kappa \end{vmatrix} = a_{11}p_2p_3\ldots p_\kappa . \tag{29} \]

Calling by \(\Delta_\kappa\) the discriminant of the form \(2\varphi\) with respect to all its \(k\) arguments \(\alpha_1,\alpha_2,\alpha_3\ldots\alpha_\kappa\), by \(\Delta_{\kappa-1}\) the discriminant of the same form when the argument \(\alpha_\kappa\) is set equal to zero, by \(\Delta_{\kappa-2}\) when the arguments \(\alpha_\kappa\) and \(\alpha_{\kappa-1}\) are set equal to zero, and so on, we shall have the series of relations:
\[ \begin{aligned} \Delta_\kappa &= a_{11}p_2p_3p_4\ldots p_\kappa,\\ \Delta_{\kappa-1} &= a_{11}p_2p_3p_4\ldots p_{\kappa-1},\\ \Delta_{\kappa-2} &= a_{11}p_2p_3p_4\ldots p_{\kappa-2},\\ &\cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\\ \Delta_3 &= a_{11}p_2p_3,\\ \Delta_2 &= a_{11}p_2,\\ \Delta_1 &= a_{11}, \end{aligned} \tag{30} \]
which make it possible to express the coefficients \(p_2,p_3,\ldots p_\kappa\) through the determinants \(\Delta_1,\Delta_2,\ldots \Delta_\kappa\).

With regard to the determinant \(\Delta_\kappa\), it must be noted that it cannot vanish, since it is the determinant of the quadratic form representing the live force of the system, which remains a positive quantity for any real values of its arguments and can become zero only when all the arguments simultaneously become zero; the same applies to all the remaining determinants
\(\Delta_{\kappa-1}, \Delta_{\kappa-2}, \ldots, \Delta_3, \Delta_2, \Delta_1\).

Turning now to the quadratic form \(2\psi\) and carrying out upon it a linear transformation with the aid of equations (28), we obtain its transformed form \(2\psi'(\beta_1,\beta_2,\beta_3,\ldots,\beta_\kappa)\), in which the coefficients will be the quantities \(b'_{ik}\), obtained in an elementary way from the coefficients \(b_{ik}\) and the coefficients of the substitutions (28).

Forming now the discriminant of the function \(\Phi' = s\varphi' - \psi'\), we find it equal to the following determinant:

\[ \Delta'(s)= \left| \begin{array}{ccccc} a_{11}s-b'_{11}, & -b'_{12}, & -b'_{13},\ldots, & -b'_{1\kappa}\\ -b'_{21}, & p_2s-b'_{22}, & -b'_{23},\ldots, & -b'_{2\kappa}\\ \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot & \cdot\\ -b'_{\kappa1}, & -b'_{\kappa2}, & -b'_{\kappa3},\ldots, & p_ns-b'_{\kappa\kappa} \end{array} \right|, \]

which differs from the discriminant \(\Delta(s)\) in that the unknown \(s\) occurs in it only in the diagonal elements, and therefore the character of its roots is determined more simply than that of \(\Delta(s)\).

Reality of the roots of the simplified discriminant \(\Delta'(s)\).

Let us represent the degenerate discriminant \(\Delta'(s)\) in the form of a bordered determinant, so that

\[ \Delta'(s)= \left| \begin{array}{ccccc} 1, & 0, & 0, & 0,0\\ 0 & a_{11}s-b'_{11}, & -b'_{12},\ldots & -b'_{1\kappa}\\ 0, & -b'_{21}, & p_2s-b'_{22},\ldots & -b'_{2\kappa}\\ \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot & \cdot\\ 0, & -b'_{\kappa1}, & -b'_{\kappa2},\ldots & p_\kappa s-b'_{\kappa\kappa} \end{array} \right| \]

and denote the determinants obtained from it by successively deleting one row (from below) and one column (from the right) by

\[ \Delta'_{\kappa-1},\ \Delta'_{\kappa-2},\ \Delta'_{\kappa-3},\ldots,\Delta'_3,\ \Delta'_2,\ \Delta'_1,\ \Delta'_0, \]

so that for some \(\Delta'_{j+1}\) we have the expression:

\[ \Delta'_{j+1}(s)= \left| \begin{array}{ccccc} 1, & 0, & \ldots, & 0\\ 0 & a_{11}s-b'_{11}, & -b'_{12} & \ldots b'_{1,j+1}\\ 0, & -b'_{21}, & p_2s-b'_{22} & \ldots\\ \cdot & \cdot & \cdot & \cdot\\ 0, & -b'_{j+1,1} & \ldots & p'_{j+1}s-b'_{j+1,j+1} \end{array} \right|. \]

\[ = \left| \begin{array}{ccccc} 1, & 0, & 0, & \ldots & 0\\ 0, & u_{11}, & u_{12}, & \ldots & u_{1,j+1}\\ 0, & u_{21}, & u_{22}, & \ldots & u_{2,j+1}\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ 0, & u_{j+1,1}, & u_{j+1,2}, & \ldots & u_{j+1,j+1} \end{array} \right|, \tag{31} \]

where, for brevity, we have put

\[ u_{jj}=p_j s-b_{jj}', \quad \text{and} \quad u_{ij}=-b_{ij}', \quad \text{and} \quad \Delta_0'=1. \]

From the manner in which the determinants \(\Delta_j'(s)\) \((j=0,1,2,3,\ldots,k)\) are formed, the following evident properties of them follow:

1) Each determinant \(\Delta_j'(s)\) is an integral polynomial arranged in descending powers of \(s\);

2) The leading term of each determinant \(\Delta_j'(s)\) has as its coefficient the determinant \(\Delta_j\), which follows directly from formula (30).

In addition to these immediately evident properties, we shall obtain still other properties of the same determinants, if we multiply the determinant \(\Delta_{j+1}'(s)\) by the determinant of order 2

\[ R_2= \left| \begin{array}{cc} M_{jj}, & M_{j,j+1}\\ M_{j+1,j}, & M_{j+1,j+1} \end{array} \right|, \]

composed of the minors \(M_{jj}\) of the determinant \(\Delta_{j+1}'(s)\) corresponding to its elements with the same indices.

For convenience in carrying out the multiplication, we shall give to the determinant of order 2 \(R_2\) the form of a determinant of order \((j+1)\):

\[ R_2= \left| \begin{array}{cccccc} 1, & 0, & 0, & \ldots 0, & 0, & 0\\ 0, & 1, & 0, & \ldots 0, & 0, & 0\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot & \cdot 1, & 0, & 0\\ M_{j1}, & M_{j2}, & M_{j3}, & \ldots M_{j,j-1}, & M_{jj}, & M_{j,j+1}\\ M_{j+1,1}, & M_{j+1,2}, & M_{j+1,3}, & \ldots M_{j+1,j-1}, & M_{j+1,j}, & M_{j+1,j+1} \end{array} \right|, \]

and after multiplication we shall have the following result:

\[ \Delta_{j+1}'(s)R_2= \left| \begin{array}{cccccc} u_{11}, & u_{12}, & u_{13}, & \ldots u_{1,j-1}, & u_{1j}, & u_{1,j+1}\\ u_{21}, & u_{22}, & u_{23}, & \ldots u_{2,j-1}, & u_{2j}, & u_{2,j+1}\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ u_{j-1,1}, & u_{j-1,2}, & u_{j-1,3}, & \ldots u_{j-1,j-1}, & u_{j-1,j}, & u_{j-1,j+1}\\ 0, & 0, & 0, & \ldots 0, & \Delta_{j+1}', & 0\\ 0, & 0, & 0, & \ldots 0, & 0, & \Delta_{j+1}' \end{array} \right| = \Delta_{j-1}'(s)\Delta_{j+1}'{}^{\,2}(s). \]

and, after reduction by \(\Delta'_{j+1}(s)\), we obtain the formula:

\[ R_2=\Delta'_{j-1}(s)\Delta'_{j+1}(s), \]

which may also be written in the following form:

\[ \tag{32} \Delta'_{j+i}\Delta'_{j-1}=M_{jj}\Delta'_j-M_{j,j+1}M_{j+1,j} \]

By the symmetry of the determinant \(\Delta'_{j+1}(s)\) and of its minors, the product \(M_{j,j+1}\cdot M_{j+1,j}=M^2_{j,j+1}\), i.e., is a quantity positive for any real value of \(s\). Therefore, if we find in some way a real root \(s=s_j\) of the equation \(\Delta'_j(s)=0\) and substitute this root \(s_j\) for \(s\) in formula (32), we obtain the following equality:

\[ \tag{33} \Delta'_{j+1}(s_j)\cdot \Delta'_{j-1}(s_j)=-[M_{j,j+1}(s_j)]^2, \]

which shows that the expressions \(\Delta'_{j+1}(s_j)\) and \(\Delta'_{j-1}(s_j)\) have different signs.

This property belongs to any three consecutive functions and may be formulated as follows: the real roots \(s_j\) of any determinant \(\Delta'_j\), when substituted into the determinants adjacent to it on the right and on the left, \(\Delta'_{j-1}\) and \(\Delta'_{j+1}\), give results with opposite signs.

From this property, relating to any three neighboring determinants, there follows a very important consequence concerning all the determinants, namely that the number of variations of signs in the sequence of determinants

\[ \tag{34} \Delta'_{\kappa}(s),\ \Delta'_{\kappa-1}(s),\ \Delta'_{\kappa-2}(s),\ldots,\Delta'_3(s),\ \Delta'_2(s),\ \Delta'_1(s),\ \Delta'_0 \]

when substituting into them, in place of \(s\), first \(s_j-\varepsilon\), and then \(s_j+\varepsilon\), where \(s_j\) is one of the real roots of the determinant \(\Delta'_j(s)\), remains unchanged.

The proof of this property is based on the continuity of all members of the sequence (34), as integral polynomials in the argument \(s\); by virtue of this continuity, under any change of \(s\) within an infinitely small interval from \(s_j-\varepsilon\) to \(s_j+\varepsilon\), all members of the sequence (34), except \(\Delta'_j(s)\), retain their signs, while \(\Delta'_j(s)\), in passing through the root, changes sign from \(+\) to \(-\), or conversely.

Taking into account that for \(s=s_j\) the signs of \(\Delta'_{j+1}(s)\) and \(\Delta'_{j-1}(s)\) are different and remain so throughout the infinitely small inter—

varying \((s_j-\varepsilon,\ s_j+\varepsilon)\), we shall have the following possible table of signs for three neighboring terms of the series (34):

	\(\Delta'_{j-1}(s)\)	\(\Delta_j(s)\)	\(\Delta_{j+1}(s)\)
\(s_j-\varepsilon\)	\(\pm\)	\(\pm\)	\(\mp\)
\(s_j+\varepsilon\)	\(\pm\)	\(\mp\)	\(\mp\)

From this table it is seen that, for any possible combination of signs in the three functions capable of introducing a change in the number of variations, this change does not occur.

It is evident without proof that no change in the number of variations of signs will occur also in the case when the real root \(s_j\) makes vanish, besides \(\Delta'_j(s)\), other terms of the series (34), provided only that these terms do not immediately follow one another.

Indeed, the properties of the middle terms of the series (34) indicated above lead to the conclusion that a change in the number of variations of signs among the terms of the series (34) occurs only when \(s\) passes through some real root of the determinant \(\Delta'(s)\).

We shall establish a complete analogy of the series (34) with Sturm’s series of functions, used in separating the real roots of algebraic equations, if we prove that the consecutive terms of the series \(\Delta_{j+1}(s)\) and \(\Delta'_j(s)\) have different signs for the values \(s_{j+1}-\varepsilon\), and the same signs for the values \(s_{j+1}+\varepsilon\), where \(s_{j+1}\) denotes one of the real roots of the determinant \(\Delta'_{j+1}(s)\) which does not make the determinant \(\Delta'_j(s)\) vanish.

The proof follows from the fact that the derivative of the determinant \(\Delta'_{j+1}(s)\) with respect to \(s\)

\[ \frac{d\Delta'_{j+1}}{ds} = a_{11}\frac{\partial \Delta'_{j+1}}{\partial u_{11}} + p_2\frac{\partial \Delta'_{j+1}}{\partial u_{22}} +\cdots+ p_{j+1}\frac{\partial \Delta'_{j+1}}{\partial u_{j+1\,j+1}} \]

\[ = a_{11}M_{11}+p_2M_{22}+\cdots+p_{j+1}\Delta'_j(s) \]

for \(s=s_{j+1}\) represents a sum of terms of one sign with \(\Delta'_j(s_{j+1})\), since all the factors \(a_{11}, p_2, p_3,\ldots,p_{j+1}\) are positive quantities (as was noted above), and all the minors \(M_{ii}(s_{j+1})\) have the same sign as \(\Delta'_j(s_{j+1})\), as is seen from formulas (32) and

analogous to it, for \(\Delta'_{j+1}(s_{j+1})=0\), which give the products

\[ \left[M_{jj}\Delta'_j\right]_{s=s_{j+1}} = \left[M_{j,j+1}\right]^2_{s=s_{j+1}}. \]

Therefore, when

\[ \Delta'_j(s_{j+1})>0,\qquad \left[\frac{d\Delta'_{j+1}}{ds}\right]_{s=s_{j+1}}>0, \]

the function \(\Delta'_{j+1}\), in passing through the root \(s_{j+1}\), increases; on the contrary, when

\[ \Delta'_j(s_{j+1})<0,\qquad \left[\frac{d\Delta'_{j+1}}{ds}\right]_{s=s_{j+1}}<0 \]

the function \(\Delta'_{j+1}\), in passing through the root \(s_{j+1}\), decreases; in short, we have the following table of signs:

	\(\Delta'_{j+1}(s)\)	\(\Delta'_j(s)\)
\(s_{j+1}-\varepsilon\)	\(\mp\)	\(\pm\),
\(s_{j+1}+\varepsilon\)	\(\pm\)	\(\pm\)

which shows that before passing through any real root of the determinant \(\Delta'_{j+1}(s)\), the signs of \(\Delta'_{j+1}\) and \(\Delta'_j(s)\) are different, while after the passage they become the same, and there always occurs the loss of one variation of signs.

On the number of real roots of the determinant \(\Delta'_{j+1}(s)\) inside a finite interval from \(A\) to \(B\).

Usually, the passage from an infinitely small interval containing one real root of the equation \(\Delta'_{j+1}(s)=0\) to an interval containing several roots is passed over in silence. Following Sturm’s example, after proving that the loss of variation of signs in the sequence

\[ \Delta'_{j+1},\ \Delta'_j \ldots \Delta'_1,\ \Delta'_0 \]

always occurs between the first two terms of this sequence as a result of the passage of \(s\) through a real root of the equation \(\Delta'_{j+1}(s)=0\), the conclusion is drawn that the difference between the numbers of variations when the values \(A\), then \(B\), are substituted for \(s\) in the terms of the sequence is equal to the number of real roots of the equation \(\Delta'_{j+1}(s)=0\) between these two-

numbers. (See, for example, Encyclopädie der Math. Wiss. I Bd, S. 417, or Weber, Lehrbuch d. Algebra, I Bd, S. 304).

With such a derivation it remains unclear in what way the losses of variations, always assigned to the first pair of determinants (or Sturm functions), are summed when the quantity \(s\) passes through the finite interval from \(A\) to \(B\).

Such an ambiguity is removed if one notes that the change of \(s\) from \(s'_{j+1}-\varepsilon\) to \(s'_{j+1}+\varepsilon\)—where \(s_{j+1}\) and \(s'_{j+1}\) are two neighboring real roots of the determinant \(\Delta_{j+1}(s)\)—causes the loss of one variation, and this loss can be attributed only to a redistribution of signs among all the terms of the series \(\Delta'_{j+1}, \Delta'_j \ldots \Delta'_0\).

This necessarily follows from the fact that the determinants \(\Delta'_{j+1}\) and \(\Delta'_j\), both for \(s=s'_{j+1}-\varepsilon\) and for \(s=s'_{j+1}+\varepsilon\), must have signs opposite to one another; consequently, under either substitution the first pair of determinants gives one variation of signs, and therefore cannot cause the loss of one variation.

Having established, for the change of \(s\) from \(s'_{j+1}-\varepsilon\) to \(s'_{j+1}+\varepsilon\), the fact of a general rearrangement of signs connected with the loss of one variation, we shall readily understand that every new enlargement of the interval of variation of \(s\), connected with passage through the 2nd real root of the equation \(\Delta_{j+1}(s)=0\), causes a new general distribution of signs, with the loss of one more variation, and so on.

Thus the change of \(s\) from \(A\) to \(B\) will produce in the series of determinants \(\Delta_{j+1}(s), \Delta'_j \ldots \Delta'_0\) a redistribution of signs with the loss of as many variations as there are real roots of the equation \(\Delta_{j+1}(s)=0\) lying inside this interval.

Arrangement of the real roots of two neighboring determinants \(\Delta'_{j+1}\) and \(\Delta'_j\) with respect to one another.

Since two neighboring determinants \(\Delta'_{j+1}\) and \(\Delta'_j\) must have the same signs after passage through a real root of the first of them, and different signs before this passage, and since this must take place when \(s\) passes through every real root, each interval between two neighboring real roots of the equation \(\Delta'_{j+1}(s)=0\) will have the property that at its beginning—immediately after passing through the first of the roots \(s_{j+1}\)—the function \(\Delta'_j(s)\) will have the same sign as \(\Delta'_{j+1}\), while at the end of the interval—immediately before \(s\) passes through the following root \(s'_{j+1}\)—the sign of \(\Delta'_j(s)\) …

changes to the opposite of the sign of \(\Delta'_{j+1}(s)\), which itself remains unchanged.

In other words, between any two consecutive real roots of the function \(\Delta'_{j+1}(s)\), the following function \(\Delta'_j(s)\) changes its sign to the opposite; and this, by the continuity of the functions \(\Delta_j(s)\), denotes the presence, in the interval under consideration, of a real root of this latter function.

Thus between the real roots of the function \(\Delta'_{j+1}\) there lies at least one real root of the function \(\Delta'_j\); in exactly the same way, between the real roots of the function \(\Delta'_j\) there are situated real roots of the function \(\Delta'_{j-1}\); between the real roots of this function lie real roots of the function \(\Delta'_{j-2}\), and so on. In short, this mutual arrangement of the real roots of any two consecutive terms of the series \(\Delta'_{j+1}, \Delta'_j \ldots\) is expressed as follows: the real roots of the determinant with the larger subscript are separated from one another by the real roots of the determinant with the smaller subscript.

The reality of all the roots of the determinant \(\Delta'(s)\) and of all the determinants \(\Delta'_j(s)\) obtained from it.

On the basis of the properties set forth above, pertaining to the determinant \(\Delta'_{j+1}(s)\) with any index \(j\) from zero to \(k-1\), one can prove that all the roots of the equation \(\Delta'(s)=0\) are real. For the proof it is only necessary to show that, upon substituting first \(-\infty\) and then \(+\infty\) for \(s\), \(k\) variations of signs are lost in the series of functions
\[ \Delta_k=\Delta',\ \Delta_{k-1},\ldots,\Delta_1. \]
This follows immediately from the remark made above, that all the coefficients of \(s\) in the diagonal terms of the determinant, i.e. \(a_{11},p_2,p_3\ldots p_k\), are positive, and consequently the coefficient of the highest power of \(s\) in the expansion of the determinant \(\Delta'(s)\), equal to the product
\[ a_{11}p_2p_3\ldots p_k \]
cannot be negative. Therefore the signs of the determinants
\[ \Delta'_j(s)\quad (j=1,2,3\ldots k) \]
for \(s=-\infty\), in the present case determined by the signs of their leading terms, will alternately be positive and negative, depending on the exponent of \(s\). Thus the substitution \(s=-\infty\) gives \(k\) variations of signs; the substitution \(s=+\infty\), however, gives not a single variation; consequently, between \(-\infty\) and \(+\infty\) there ...

include all \(k\) roots of the equation \(\Delta'(s)=0\), and consequently all of them are real.

By the same method we could separately find the number of positive and the number of negative roots of the equation \(\Delta'(s)=0\), if we had the possibility, in one way or another, of determining the number of variations of signs in the sequence of determinants \(\Delta'_j(s)\) upon substituting \(s=0\). Since for \(s=0\) the determinant \(\Delta'_j(s)\) reduces to the determinant:

\[ D'= \left| \begin{array}{cccc} -b'_{11}, & -b'_{12}, & \ldots & -b'_{1j} \\ -b'_{21}, & -b'_{22}, & \ldots & -b'_{2j} \\ \cdot & \cdot & \cdot & \cdot \\ -b'_{j1}, & -b'_{j2}, & \ldots & -b'_{jj} \end{array} \right| = (-1)^j \left| \begin{array}{cccc} b'_{11}, & b'_{12}, & \ldots & b'_{1j} \\ b'_{21}, & b'_{22}, & \ldots & b'_{2j} \\ \cdot & \cdot & \cdot & \cdot \\ b'_{j1}, & b'_{j2}, & \ldots & b'_{jj} \end{array} \right| = (-1)^j D_j, \]

the final resolution of the question depends on the signs of

\[ D_j\;(j=1,2,3\ldots k), \]

which are the determinants or discriminants of the quadratic form \(\psi\), or \(V-V_0\), and of the forms obtained from them when one, two, three, etc. of their arguments are set equal to zero. A little below we shall have occasion to return to this question; for the moment we shall confine ourselves to the observation that among the real roots of the equation \(\Delta'(s)=0\) there are no roots equal to zero.

This follows from the fact that, for \(s=0\), \(\Delta'(s)\) becomes \(D_k\), the discriminant of the quadratic form \(\psi\), or \(V-V_0\). Since \(V\) is the potential function of the system, for which the particular value \(V_0\), in the position of equilibrium, is a minimum of its values and which therefore cannot become less than \(V_0\) for any real values of its arguments, it is clear that the difference \(V-V_0\), or \(\psi\), is always positive, i.e. the form \(\psi\) is what is called a forma definita, and its determinant \(D_k\) cannot be equal to zero.

Passage from the differential equations to integrals.

Having established the reality of all the roots of the harmonizing determinant \(\Delta'(s)\), and considering these roots as computed (exactly or to the desired approximation), we can now pass from the initial Lagrange equations of motion to equations of harmonic type (14), obtained from the former by combining them.

These equations (14) differ from one another only in the values of the coefficient \(s\); therefore the integration of all of them is performed indepe...

depend on one another according to one and the same pattern. The number of integrals obtained will be equal to the number of distinct roots of the equation \(\Delta'(s)=0\), and consequently may either be equal to the number of independent coordinates of the system, or be less than this number, depending on whether all the roots of the equation \(\Delta'(s)=0\) are distinct, or whether it also has multiple roots.

The necessity of distinguishing these two cases becomes especially clear at the very first attempt to determine each of the independent Lagrangian coordinates separately: in the first case, from \(k\) distinct integrals one directly finds all \(k\) independent coordinates as functions of time and of \(2k\) arbitrary constants of integration; in the second case, the number of independent coordinates exceeds the number of distinct integrals at our disposal, and these coordinates cannot be determined with the same degree of generality as in the first case until we supplement the system of integrals obtained by new equations of the same type, but independent of them.

Determination of the independent coordinates when all the roots of the equation
\[ \Delta'(s)=0 \]
are simple.

The solution of the problem posed is divided into two parts: the first comprises the construction and integration of equations of harmonic type; the second, the determination, by means of the integrals obtained, of the separate coordinates of the system.

Proceeding to the first part of the problem, we shall start from equation (14). Assuming that in this equation the multipliers \(\alpha_i\) are determined according to formulas \((a)\), we obtain from it the second-order differential equation:

\[ \tag{g} \frac{d^2}{dt^2}\sum_{i=1}^{i=k}\alpha_{ih}Q_i+\sum_{i=1}^{i=k}\alpha_{ih}V_i=0, \]

in which, by virtue of formula \((s)\),

\[ \sum \alpha_{ih}V_i=\sum \alpha_{ih}Q_i. \]

Putting, for brevity,

\[ \sum \alpha_{ih}Q_i=U_h, \]

we shall give equation \((g)\) the form of an equation of harmonic type:

\[ \frac{d^{2}U_h}{dt^{2}}+s_h U_h=0. \tag{h} \]

According to the number of real roots \(s_h\), we shall obtain \(k\) such equations. The integration of these equations is carried out very simply and leads to a system of \(k\) integrals of the form:

\[ U_h=A_h e^{\sqrt{-s_h}\,t}+B_h e^{-\sqrt{-s_h}\,t} =A_h e^{i t n_h}+B_h e^{-i t n_h} \tag{l} \]

where \(n_h=\sqrt{s_h}\), and \(A_h\) and \(B_h\) are arbitrary constants of integration. According as \(s_h\) is a positive or negative quantity, the integrals \(U_h\) will be expressed either by circular or by hyperbolic functions. Without dwelling on this for the present, we shall pass to the second part of the question.

Determination of the independent coordinates \(q_i\).

The direct dependence of the coordinates \(q_i\) on \(U_h\) can be established in two different ways: either 1) by first computing the polynomials \(U_h\) as explicit functions of \(q_i\), or 2) by first finding expressions for the various \(Q_i\) as functions of \(U_h\), and then solving the resulting equations with respect to \(q_i\).

Choosing the first way, we note at once that the coefficients of the coordinates \(q_i\) in the polynomials \(U_h\) will obey, in their formation, the same law to which, in their formation, the various elements of a determinant are subject, that determinant being the product of two other determinants of one and the same order. In the present case such determinants will be: 1) the determinant \(\Delta\) of the coefficients \(a_{ik}\) of the definite (forma definita) quadratic form \(Q\), and 2) the determinant of the coefficients \(\alpha_{ih}\), which we shall denote by \(\Delta(\alpha)\).

Therefore the question of the solvability or non-solvability of the system \((k)\) with respect to the variables \(q_i\), by the first method, is reduced to the question whether the product of the determinants \(\Delta \cdot \Delta(\alpha)\) will or will not be equal to zero; and since \(\Delta\) cannot be equal to zero, everything will depend in the end on the determinant \(\Delta(\alpha)\). On this same determinant will depend the success of the solution when choosing the second way as well. Relying on the fact that a determinant becomes zero 1) when all the elements of one and the same row or column are equal to zero, or 2) when all the elements of two different rows or columns are proportional, we could prove that

\(\Delta(\alpha)\) is different from zero, to prove that neither equality to zero nor proportionality in the sense indicated above takes place.

Although giving a proof of this kind presents no difficulty, we prefer to replace the simple proof of the possibility of computing \(q_i(i=1,2,3,\ldots k)\) by the establishment of formulas by which the \(q_i\) are computed directly as functions of the \(U_h\).

The path leading to these formulas is indicated, to a certain extent, by the very form of the equations:

\[ \frac{d^2 U_h}{d t^2}+s_h U_h=0\quad (h=1,2,3,\ldots k), \]

if one looks at them from a new point of view.

Until now we have regarded these equations as the result of combining the original Lagrange equations of the form:

\[ \frac{d^2}{d t^2}\left(\frac{\partial Q}{\partial q_i}\right)+\frac{\partial V}{\partial q_i}=0\quad (i=1,2,3\ldots k); \]

now we shall agree to regard the combined equations as original Lagrange equations in which the role of independent coordinates has passed from the variables \(q_i\) to new variables—the polynomials \(U_h\). From this point of view \(U_h\) and \(s_h U_h\) must be identified with the partial derivatives, with respect to the argument \(U_h\), of certain homogeneous quadratic functions \(U(U_1,U_2,U_3,\ldots U_k)\) and \(W(U_1,U_2,\ldots U_k)\), i.e. the equalities must hold:

\[ (u)\qquad \frac{\partial U}{\partial U_h}=U_h \quad \text{and} \quad \frac{\partial W}{\partial U_h}=s_h U_h\quad (h=1,2,3,\ldots k). \]

On the basis of these conditions, and also of the condition that the functions \(U\) and \(W\) must be homogeneous quadratic functions, by integration we obtain for them the following expressions:

\[ 2U=\sum_{h=1}^{h=k} U_h^2 \quad \text{and} \quad 2W=\sum_{h=1}^{h=k} s_h U_h^2. \]

By the manner of their formation, \(U\) and \(W\) are explicit functions of the variables \(U_h\) and implicit functions of the independent variables \(q_i\); moreover, knowing the connection of the polynomials \(U_h\) with the quadratic functions \(Q\) and \(V\), we shall at once note that \(U\) and \(W\) are the results of one and the same linear transformation performed on \(Q\) and \(V\), and conversely.

The investigation carried out above of the nature of the roots of the harmonizing determinant \(\Delta(s)\) frees us from the necessity of setting forth the theory of the simultaneous transformation of two quadratic forms in all its details.

Following the generally accepted procedure, we shall replace the transformation of the two quadratic forms \(Q\) and \(V\) by the transformation of one composite form \(\Phi = sQ - U\), which, for particular values of the undetermined multiplier \(s\), may coincide with each of the forms under consideration. The immediate aim of the transformation will be to annihilate, in the transformed form, the terms representing products of distinct arguments of this form by one another; and the basis of this transformation will be the decomposition into partial fractions of the algebraic fraction \(G(s):\Delta(s)\), whose denominator is the harmonizing determinant \(\Delta(s)\), while the numerator \(G(s)\) is obtained from the same determinant \(\Delta(s)\) by bordering it on the left and above with the row of quantities

\[ 0,\ \Phi_1,\ \Phi_2,\ \ldots,\ \Phi_k, \]

so that

\[ G(s)= \left| \begin{array}{ccccc} 0, & \Phi_1, & \Phi_2, & \ldots, & \Phi_k \\ \Phi_1, & a_{11}s-b_{11}, & a_{12}s-b_{12}, & \ldots, & a_{1k}s-b_{1k} \\ \Phi_2, & a_{21}s-b_{21}, & a_{22}s-b_{22}, & \ldots, & a_{2k}s-b_{2k} \\ \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot \\ \Phi_k, & a_{k1}s-b_{k1}, & a_{k2}s-b_{k2}, & \ldots, & a_{kk}s-b_{kk} \end{array} \right| \tag{g} \]

Regarding \(\Phi_1,\Phi_2,\ldots,\Phi_k\) temporarily as quite arbitrary quantities, we decompose the fraction \(G(s):\Delta(s)\), according to the ordinary rules, into a sum of partial fractions and obtain:

\[ \frac{G(s)}{\Delta(s)} = \sum_{i=1}^{i=k}\frac{A_i}{s-s_i}. \tag{f} \]

Since all the roots \(s_i\) are assumed to be distinct from one another, the number of partial fractions will be equal to the number of independent variables of the form \(\Phi\), and the coefficients \(A_i\) will be determined by the formula:

\[ A_i=\left\{G(s):\frac{d\Delta(s)}{ds}\right\}_{s=s_i}. \tag{\(A_i\)} \]

Restricting ourselves, as regards the denominator of the coefficient \(A_i\), to the brief remark that it cannot vanish either for ...

some \(s_i\), we shall transform the numerator of the same coefficient, i.e. the determinant \(G(s_i)\), which has the form:

\[ (g_i)\qquad G(s_i)= \left| \begin{array}{ccccc} 0, & \Phi_1, & \Phi_2, & \ldots & \Phi_k\\ \Phi_1, & a_{11}s_i-b_{11}, & a_{12}s_i-b_{12}, & \ldots & a_{1k}s_i-b_{1k}\\ \Phi_2, & a_{21}s_i-b_{21}, & a_{22}s_i-b_{22}, & \ldots & a_{2k}s_i-b_{2k}\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ \Phi_k, & a_{k1}s_i-b_{k1}, & a_{k2}s_i-b_{k2}, & \ldots & a_{kk}s_i-b_{kk} \end{array} \right|. \]

Since this formula \((g_i)\) is obtained from formula \((g)\) for arbitrary values \(\Phi_1,\Phi_2,\ldots,\Phi_k\), we may apply it also to the particular case in which the quantities \(\Phi_i\) are replaced by their actual values, i.e. when

\[ \Phi_i=sQ_i-V_i\quad (i=1,2,3,\ldots,k). \]

If, after such replacement of all \(\Phi_i\) by their values, we multiply the columns of the determinant \((g_i)\), beginning with the second, successively by \(q_1,q_2,\ldots,q_k\), and subtract the results of the multiplication from the first column, then, while leaving the value of the determinant unchanged, we shall give it the following form:

\[ G(s_i)= \left| \begin{array}{ccccc} -\displaystyle\sum_{1}^{k}\Phi_i q_i, & \Phi_1, & \Phi_2, & \ldots & \Phi_k\\ (s-s_i)Q_1, & a_{11}s_i-b_{11}, & a_{12}s_i-b_{12}, & \ldots & a_{1k}s_i-b_{1k}\\ (s-s_i)Q_2, & a_{21}s_i-b_{21}, & a_{22}s_i-b_{22}, & \ldots & a_{2k}s_i-b_{2k}\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ (s-s_i)Q_k, & a_{k1}s_i-b_{k1}, & a_{k2}s_i-b_{k2}, & \ldots & a_{kk}s_i-b_{kk} \end{array} \right|. \]

Then, multiplying the rows of the determinant thus obtained, except the first, successively by \(q_1,q_2,\ldots,q_k\), and subtracting from the first, we obtain a new form of the same determinant:

\[ G(s_i)= \left| \begin{array}{ccccc} -2\Phi-(s-s_i)\displaystyle\sum_{1}^{k}Q_iq_i, & (s-s_i)Q_1, & (s-s_i)Q_2, & \ldots & (s-s_i)Q_k\\ 0, & +(s-s_i)Q_1, & a_{11}s_i-b_{11}, & a_{12}s_i-b_{12}, & \ldots\, a_{1k}s_i-b_{1k}\\ 0, & +(s-s_i)Q_2, & a_{21}s_i-b_{21}, & a_{22}s_i-b_{22}, & \ldots\, a_{2k}s_i-b_{2}\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ 0, & +(s-s_i)Q_k, & a_{k1}s_i-b_{k1}, & a_{k2}s_i-b_{k2}, & \ldots\, a_{kk}s_i-b_{kk} \end{array} \right|. \]

\[ \begin{aligned} &= \left| \begin{array}{ccccc} -2\Phi-2(s-s_i)Q, & (s-s_i)Q_1, & (s-s_i)Q_2, & \cdots, & (s-s_i)Q_k\\ 0, & +(s-s_i)Q_1, & a_{11}s_i-b_{11}, & a_{12}s_i-b_{12}, & a_{1k}s_i-b_{1k}\\ 0, & +(s-s_i)Q_2, & a_{21}s_i-b_{21}, & \cdots, & a_{2k}s_i-b_{2k}\\ \cdots & \cdots & \cdots & \cdots & \cdots\\ 0, & +(s-s_i)Q_k, & a_{k1}s_i-b_{k1}, & \cdots, & a_{kk}s_i-b_{kk} \end{array} \right| \\[1em] &=2\Phi\,\Delta(s_i)+(s-s_i)^2 \left| \begin{array}{ccccc} -\dfrac{2Q}{s-s_i}+0, & Q_1, & Q_2, & \cdots, & Q_k\\ Q_1+0, & a_{11}s_i-b_{11}, & a_{12}s_i-b_{12}, & \cdots, & a_{1k}s_i-b_{1k}\\ Q_2+0, & a_{21}s_i-b_{21}, & a_{22}s_i-b_{22}, & \cdots, & a_{2k}s_i-b_{2k}\\ \cdots & \cdots & \cdots & \cdots & \cdots\\ Q_k+0, & a_{k1}s_i-b_{k1}, & a_{k2}s_i-b_{k2}, & \cdots, & a_{kk}s_i-b_{kk} \end{array} \right|. \end{aligned} \]

or, since \(\Delta(s_i)=0\),

\[ \begin{aligned} G(s_i) &= \left| \begin{array}{ccccc} -\dfrac{2Q}{s-s_i}+0, & Q_1, & Q_2, & \cdots, & Q_k\\ 0+Q_1, & a_{11}s_i-b_{11}, & a_{12}s_i-b_{12}, & \cdots, & a_{1k}s_i-b_{1k}\\ 0+Q_2, & a_{21}s_i-b_{21}, & \cdots, & \cdots, & a_{2k}s_i-b_{2k}\\ \cdots & \cdots & \cdots & \cdots & \cdots\\ 0+Q_k, & a_{k1}s_i-b_{k1}, & \cdots, & \cdots, & a_{kk}s_i-b_{kk} \end{array} \right|\,(s-s_i)^2 \\[1em] &=(s-s_i)^2 \left| \begin{array}{ccccc} 0, & Q_1, & Q_2, & \cdots, & Q_k\\ Q_1, & a_{11}s_i-b_{11}, & a_{12}s_i-b_{12}, & \cdots, & a_{1k}s_i-b_{1k}\\ Q_2, & a_{21}s_i-b_{21}, & a_{22}s_i-b_{22}, & \cdots, & a_{2k}s_i-b_{2k}\\ \cdots & \cdots & \cdots & \cdots & \cdots\\ Q_k, & a_{k1}s_i-b_{k1}, & a_{k2}s_i-b_{k2}, & \cdots, & a_{kk}s_i-b_{kk} \end{array} \right| \qquad (G(s_i)). \end{aligned} \]

Making the same substitution of the quantities \(\Phi_1,\Phi_2,\ldots,\Phi_k\) by

\[ Q_1s-V_1,\quad Q_2s-V_2,\ldots,Q_ks-V_k \]

in the determinant \(G(s)\), after a series of the same transformations as those performed on the determinant \(G(s_i)\), we obtain the following formula:

\[ \begin{aligned} G(s) &= \left| \begin{array}{ccccc} 0, & \Phi_1, & \Phi_2, \ldots & \Phi_k\\ \Phi_1, & a_{11}s-b_{11}, \ldots & a_{1k}s-b_{1k}\\ \Phi_2, & a_{21}s-b_{21}, \ldots & a_{2k}s-b_{2k}\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ \Phi_k, & a_{k1}s-b_{k1}, \ldots & a_{kk}s-b_{kk} \end{array} \right| \\ &= \left| \begin{array}{ccccc} -\displaystyle\sum_{1}^{k}\Phi_i q_i, & \Phi_1, & \Phi_2, \ldots & \Phi_k\\ 0, & a_{11}s-b_{11}, \ldots & a_{1k}s-b_{1k}\\ 0, & a_{21}s-b_{21}, \ldots & a_{2k}s-b_{2k}\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ 0, & a_{k1}s-b_{k1}, \ldots & a_{kk}s-b_{kk} \end{array} \right| \\ &= -\sum_{i=1}^{i=k}\Phi_i q_i\cdot \Delta(s)=-2\Phi\cdot\Delta(s) \end{aligned} \tag{G(s)} \]

Substituting this value of \(G(s)\) into the expansion of the fraction \(G(s):\Delta(s)\), we find that:

\[ -2\Phi=\sum_{i=1}^{i=k}\frac{A_i}{s-s_i} =+\sum_{i=1}^{i=k} \left\{ \frac{(s-s_i)^2}{(s-s_i)\left(\dfrac{d\Delta}{ds}\right)_{s=s_i}} \left| \begin{array}{ccccc} 0, & Q_1, & Q_2,\ldots & Q_k\\ Q_1, & a_{11}s_i-b_{11}, \ldots & a_{1k}s_i-b_{1k}\\ Q_2, & a_{21}s_i-b_{21}, \ldots & a_{2k}s_i-b_{2k}\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ Q_k, & \ldots & \ldots & a_{kk}s_i-b_{kk} \end{array} \right| \right\}, \]

or, after reduction:

\[ 2\Phi = -\sum_{i=1}^{i=k} \frac{s-s_i} {\left(\dfrac{d\Delta}{ds}\right)_{s=s_i}} \left| \begin{array}{ccccc} 0, & Q_1, & Q_2,\ldots & Q_k\\ Q_1, & a_{11}s_i-b_{11}, \ldots & a_{1k}s_i-b_{1k}\\ Q_2, & a_{21}s_i-b_{21}, \ldots & a_{2k}s_i-b_{2k}\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ Q_k, & a_{k1}s_i-b_{k1}, \ldots & a_{kk}s_i-b_{kk} \end{array} \right|. \]

The bordered determinant entering into the last formula for \(2\Phi\), when expanded with respect to the elements of the border, will have the form of a homogeneous quadratic form in the arguments \(Q_1,Q_2,\ldots Q_k\); the coefficients of this form will be the minors of the determinant \(\Delta(s_i)\), conjugate to those elements of it which stand at the intersection of the vertical and horizontal lines containing those \(Q\)'s which enter into the given term of the quadratic form.

Using for the designation of these minors the former symbols \(M\) with the corresponding indices, we shall write the expansion of the determinant just mentioned in the following form:

\[ \begin{vmatrix} 0, & Q_1, & Q_2, & \ldots & Q_k \\ Q_1, & a_{11}s_i-b_{11}, & a_{12}s_i-b_{12}, & \ldots & \\ Q_2, & a_{21}s_i-b_{21}, & a_{22}s_i-b_{22}, & \ldots & \\ \cdot & \cdot & \cdot & \cdot & \cdot \\ Q_k, & a_{k1}s_i-b_{k1}, & a_{k2}s_i-b_{k2}, & \ldots & a_{kk}s_i-b_{kk} \end{vmatrix} \]

\[ = -\bigl(M_{11}(s_i)Q_1^2+2M_{12}(s_i)Q_1Q_2+\cdots+M_{kk}(s_i)Q_k^2\bigr) = -\sum_{e j} M_{ej}(s_i)Q_eQ_j \tag{q} \]

But, according to the remark made above, when the determinant \(\Delta(s_i)\) is identically equal to zero, the minors of the elements of any row will be proportional to the minors of the elements of every other row, so that for any two indices \(j\) and \(k\) we have:

\[ \frac{M_{jj}}{M_{jk}}=\frac{M_{jk}}{M_{kk}}; \]

by virtue of the symmetry of the determinant \(\Delta(s_i)\),

\[ M_{jk}(s_i)=M_{kj}(s_i), \]

therefore the minors of the given determinant satisfy the following equalities:

\[ M_{jj}(s_i)\cdot M_{kk}(s_i)=M_{jk}^{\,2}(s_i), \quad \text{or} \quad M_{kj}(s_i)=\sqrt{M_{jj}}\cdot\sqrt{M_{kk}}. \]

If, using these equalities, one expands all the coefficients of the quadratic form \((q)\), then this form becomes a complete square of the following form:

\[ \sum_{e j} M_{je}(s_i)\cdot Q_eQ_j = \sum_{e j}\sqrt{M_{ee}(s_i)}\cdot\sqrt{M_{jj}(s_i)}\,Q_e\cdot Q_j \]

\[ = \bigl(\sqrt{M_{11}(s_i)}\,Q_1+\sqrt{M_{22}(s_i)}\,Q_2+\cdots+\sqrt{M_{kk}(s_i)}\,Q_k\bigr)^2. \]

Substituting the decomposition obtained into the formula for \(2\Phi\), we shall have:

\[ 2\Phi = \sum_{i=1}^{i=k} \frac{(s-s_i)} {\left(\dfrac{d\Delta}{ds}\right)_{s=s_i}} \cdot \bigl( \sqrt{M_{11}(s_i)}\,Q_1+ \sqrt{M_{22}(s_i)}\,Q_2+\cdots+ \sqrt{M_{kk}(s_i)}\,Q_k \bigr)^2 \]

or

\[ 2\Phi=\sum_{i=1}^{i=k}(s-s_i)\cdot \left\{ \sqrt{\frac{M_{11}(s_i)}{\left(\dfrac{d\Delta}{ds}\right)_{s_i}}}\,Q_1 + \sqrt{\frac{M_{22}(s_i)}{\left(\dfrac{d\Delta}{ds}\right)_{s_i}}}\,Q_2 +\cdots+ \sqrt{\frac{M_{kk}(s_i)}{\left(\dfrac{d\Delta}{ds}\right)_{s_i}}}\,Q_k \right\}^{2} \]

\[ =\sum_{i=1}^{i=k}(s-s_i)\{\,{}^{i}a_1Q_1+{}^{i}a_2Q_2+\cdots+{}^{i}a_kQ_k\,\}^{2}; \]

where, following Jacobi’s example, by \({}^{i}a_j\) are denoted the coefficients of \(Q_j\), i.e.

\[ \left[ \sqrt{\frac{M_{jj}}{\left(\dfrac{d\Delta}{ds}\right)}} \right]_{s=s_i}. \]

Calling temporarily by \(\omega_i\) the linear expressions

\[ {}^{i}a_1Q_1+{}^{i}a_2Q_2+{}^{i}a_3Q_3+\cdots+{}^{i}a_kQ_k, \]

we shall write \(2\Phi\) in the form:

\[ 2\Phi=\sum_{i=1}^{i=k}(s-s_i)\omega_i^2, \]

or, decomposing the left- and right-hand sides into terms, we shall have:

\[ 2\Phi=2sQ-2V=s\sum_{i=k}^{i=k}\omega_i^2-\sum_{i=1}^{i=k}s_i\omega_i^2. \]

For arbitrary \(s\), such an equality decomposes into two equalities:

\[ 2Q=\sum_{i=1}^{i=k}\omega_i^2 \quad\text{and}\quad 2V=\sum_{i=1}^{i=k}s_i\omega_i^2. \]

The results obtained, compared with the quadratic functions \(2U\) and \(2W\), show that

\[ 2U=\sum_{1}^{\kappa}\omega_i^2 \quad\text{and}\quad 2W=\sum_{1}^{\kappa}s_i\omega_i^2, \]

and

\[ U_h=\omega_h={}^{h}a_1Q+{}^{h}a_2Q_2+\cdots+{}^{h}a_kQ_k.\quad (h=1,2,3,\ldots k). \]

The last formula will be useful in solving our main problem: to determine the independent variables \(q_e\) as functions of \(U_h\).

Directly, the variables \(q_e\) are connected with the polynomials \(\Phi_j\) \((j=1,2,3,\ldots k)\) by a system of linear equations of the form:

\[ (a_{j1}s-b_{j1})q_1+(a_{j2}s-b_{j2})q_2+\cdots+(a_{je}s-b_{je})q_e+\cdots+(a_{jk}s-b_{jk})q_k =\Phi_j \quad (j=1,2,\ldots k). \]

Considering \(\Phi_j\) as new variables, we solve this system with respect to the old variables \(q_e\), and shall have:

\[ q_e=\frac{1}{\Delta(s)}\sum_{j=1}^{j=k}M_{je}(s)\Phi_j \tag{\(q_e\)} \]

Let us decompose, as we did earlier, the algebraic fraction forming the right-hand side of the equality \((q_e)\) into a sum of partial fractions, and represent \(q_e\) in the form:

\[ q_e=\sum_{i=1}^{i=k}\frac{B_j}{s-s_i}, \tag{B} \]

where

\[ \begin{aligned} B_j &=\frac{1}{\left(\dfrac{d\Delta}{ds}\right)_{s=s_i}} \sum_{j=1}^{j=k}M_{je}(s_i)\Phi_j =\sum_{j=1}^{j=k}\Phi_j \frac{M_{je}(s_i)}{\left(\dfrac{d\Delta}{ds}\right)_{s=s_i}} \\ &=\sum_{j=1}^{j=k}\Phi_j \frac{\sqrt{M_{jj}(s_i)}\cdot\sqrt{M_{ee}(s_i)}}{\left(\dfrac{d\Delta}{ds}\right)_{s=s_i}} \\ &=\sum_{j=1}^{j=k} a_j^{\,i} a_e^{\,i}\Phi_j . \end{aligned} \]

Introducing this transformed expression for \(B_i\) into the expression for \(q_e\), we shall have:

\[ q_e=\sum_{i=1}^{i=k}\sum_{j=1}^{j=k}\frac{a_j^{\,i}a_e^{\,i}\Phi_j}{s-s_i} =\sum_{i=1}^{i=k}a_e^{\,i}\sum_{j=1}^{j=k}\frac{a_j^{\,i}\Phi_j}{s-s_i}. \]

Noting that

\[ \frac{\Phi_j}{s-s_i} = \frac{sQ_j-V_j}{s-s_i} = \frac{Q_j(s-s_i+s_i)-V_j}{s-s_i} = Q_j+\frac{s_iQ_j-V_j}{s-s_i}, \]

we shall give to the inner sum entering the last expression for \(q_e\) the following form:

\[ (\Sigma)\qquad \sum_{j=1}^{j=\kappa} \frac{a_j^{\,i}\Phi_j}{s-s_j} = \sum_{j=1}^{j=\kappa} a_j^{\,i}Q_j + \sum_{j=1}^{j=\kappa} \frac{a_j^{\,i}(s_jQ_i-V_j)}{s-s_i} = \omega_i+ \sum_{j=1}^{j=\kappa} \frac{a_j^{\,i}(s_iQ_j-V_j)}{s-s_i}. \]

After substituting this transformed sum into the formula for \(q_e\), we obtain for \(q_e\) a new expression:

\[ q_e=\sum_{i=1}^{i=\kappa} a_e^{\,i}\omega_i + \sum_{i=1}^{i=\kappa}\sum_{j=1}^{j=\kappa} \frac{a_e^{\,i}a_j^{\,i}(s_iQ_j-V_j)}{s-s_i}. \]

It can be shown that the double sum in the formula just obtained for \(q_e\) vanishes; for this it is only necessary to replace the product of coefficients \(a_e^{\,i}a_j^{\,i}\) by

\[ M_{je}(s_i):\left(\frac{d\Delta}{ds}\right)_{s=s_i}; \]

then the expression

\[ \frac{a_e^{\,i}a_j^{\,i}(s_iQ_j-V_j)}{s-s_i} \]

is transformed into

\[ \frac{M_{je}(s_i)(s_iQ_j-V_j)} {(s-s_i)\cdot\left(\dfrac{d\Delta}{ds}\right)_{s=s_i}}, \]

whose numerator, i.e.

\[ (s_iQ_j-V_j)M_{je}(s_i) = \{q_1(a_{j1}s_i-b_{j1})+q_2(a_{j2}s_i-b_{j2})+\cdots+q_\kappa(a_{j\kappa}s_i-b_{j\kappa})\}M_{je}(s_i), \]

when summed over the index \(j\) (for one and the same \(i\)), will be equal to zero, since the coefficients of the various \(q_r\) will have the form:

\[ \sum_{j=1}^{j=\kappa}(a_{jr}s_i-b_{jr})M_{je}(s_i), \]

which for \(r\ne e\) gives zero, as the expansion of a determinant with two identical rows, and for \(r=e\) also gives zero, as the expansion of the determinant \(\Delta(s_i)\), which is identically equal to zero.

The vanishing of the inner sum entails the annihilation of the entire second term in the expression for \(q_e\), and we obtain for \(q_e\) the final expression:

\[ q_e=\sum_{i=1}^{i=k} a_e^i \omega_i=\sum_{i=1}^{i=k} a_e^i U_i, \]

whereby our problem is brought to completion.

Substitution in place of \(U_i\) of the two-term expressions

\[ A_i e^{i n_i t}+B_i e^{-i n_i t} \]

will lead to a system of integrals whose general form will be:

\[ q_e=\sum_{i=1}^{i=k} A_i a_e^i e^{i n_i t} +\sum_{i=1}^{i=k} B_i a_e^i e^{i n_i t} \qquad (i=1,2,3\ldots k). \]

The number of arbitrary constants \(A_i\) and \(B_i\) of this system will be equal to \(2k\), i.e. to twice the number of independent coordinates of the system.

Integration of Lagrange’s equations when the determinant \(\Delta(s)\) has multiple roots.

When the harmonizing determinant \(\Delta(s)\) of Lagrange’s equations, besides simple roots, has one or several multiple roots, the method of integration adopted by us in its pure form proves capable of giving only those differential equations of harmonic type which depend on the simple roots of the determinant \(\Delta(s)\). To obtain by the same method even one additional equation with the help of multiple roots proves impossible, for the reason that all minors of the determinant \(\Delta(s)\), and consequently also the harmonizing multipliers \(a\) proportional to them, for all multiple roots vanish, as we shall see below, identically to zero.

Therefore, in order to obtain the general solution of the problem with the full number of arbitrary constants, it is necessary to modify the method in such a way that it makes it possible to find the missing number of differential equations. Since in the modification of the method that will be set forth, an essential role is played by the properties of the minors of the determinant \(\Delta(s)\) having multiple roots, it is natural first of all to turn to the exposition of these properties.

Properties of minors of different orders of a harmonizing determinant having multiple roots.

If we agree to regard the appearance of a multiple root as the result of the coalescence of two, three, or in general of some number of initially distinct roots, then from the theory set forth above of the distribution of real roots among the various determinants of the series \((\Delta')\), we immediately obtain the following conclusions:

1) a double root of the determinant \(\Delta'(s)\) is at the same time a simple root of the determinant \(\Delta'_{k-1}(s)\), or of some other minor of a diagonal element of the determinant \(\Delta'(s)\);

2) a triple root of the determinant \(\Delta'(s)\) is a double root of the determinant \(\Delta'_{k-1}(s)\), and a simple root of the determinant \(\Delta'_{k-2}(s)\), or of some other diagonal minors;

3) in general, a \(\mu\)-fold root of the determinant \(\Delta'(s)\) is a root of multiplicities \((\mu-1)\)-st, \((\mu-2)\)-nd, ... and so on, respectively, for the minors \(\Delta'_{k-1}(s)\), \(\Delta'_{k-2}(s)\), ... and so on, or for some other minors of diagonal elements.

In fact, a real root of the determinant \(\Delta'_{k-1}(s)\) lies between two neighboring real roots of the determinant \(\Delta'(s)\); consequently, when these two roots approach one another until they coincide, the root of the determinant \(\Delta'_{k-1}(s)\) will also coincide with them. In exactly the same way, between \(\mu\) distinct real roots of the determinant \(\Delta'(s)\) lie \(\mu-1\) real roots of the determinant \(\Delta'_{k-1}(s)\); between \(\mu-1\) roots of this determinant lie \(\mu-2\) roots of the determinant \(\Delta'_{k-2}(s)\), and so on.

When all \(\mu\) roots of the determinant \(\Delta'(s)\) coalesce into one \(\mu\)-fold root, all the roots of the lower determinants will likewise coalesce with them and will be, respectively, \((\mu-1)\)-fold, \((\mu-2)\)-fold, and so on, roots of the determinants \(\Delta'_{k-1}(s)\), \(\Delta'_{k-2}(s)\), ... and so on.

The theorem of higher algebra stating that a multiple root of an integral rational polynomial is a root, of multiplicity one less, of the first derivative of this polynomial, makes it possible to prove that the properties we have noted of the diagonal or principal minors of the determinant \(\Delta'(s)\) belong not only to these minors alone, but in general to all minors of the very same order.

Let us first take the simplest case and prove that a double root \(s=\beta\) of the determinant \(\Delta'(s)\) will at the same time be a root of all first minors of this determinant. To prove this proposition, let us form the derivative of the determinant \(\Delta'(s)\) with respect to \(s\). Denote—

denote, for brevity, the various elements of the determinant \(\Delta'(s)\) by the letter \(U\) with the corresponding two indices, and, observing that \(s\) enters only into the diagonal terms of the determinant, we shall have, on the basis of the rules for differentiating determinants, the following formula:

\[ \frac{d\Delta'(s)}{ds} = \frac{\partial\Delta'(s)}{\partial U_{11}}\cdot\frac{dU_{11}}{ds} + \frac{\partial\Delta'(s)}{\partial U_{22}}\cdot\frac{dU_{22}}{ds} +\ldots+ \frac{\partial\Delta'(s)}{\partial U_{ii}}\cdot\frac{dU_{ii}}{ds} +\ldots \]
\[ + \frac{\partial\Delta'(s)}{\partial U_{kk}}\cdot\frac{dU_{kk}}{ds}. \tag{d} \]

By the condition that \(s=\beta\) is a double root of the determinant \(\Delta'(s)\) and a simple root of its derivative \(\dfrac{d\Delta'(s)}{ds}\), the left-hand side of equation \((d)\) becomes zero when \(\beta\) is substituted for \(s\); consequently, under the same substitution the right-hand side must also become zero, i.e. the sum

\[ \sum_{i=1}^{i=k} \left[ \frac{\partial\Delta'(s)}{\partial U_{ii}} \frac{dU_{ii}}{ds} \right]_{s=\beta} = \sum_{i=1}^{i=k} M_{ii}(\beta)\,p_i = \sum_{i=1}^{i=k} a_{ii}M_{ii}(\beta). \]

But the factors \(a_{11}, p_2,\ldots p_i\ldots p_k\), as coefficients of the squares in a positive definite quadratic function (forma definita), are all positive; likewise positive are all the partial derivatives \(\dfrac{\partial\Delta'(s)}{\partial U_{ii}}\), or the diagonal minors \(M_{ii}\), of the determinant \(\Delta'(s)\); therefore the entire right-hand side of the expression \(\dfrac{d\Delta'(s)}{ds}\) consists of a sum of positive terms only, and its vanishing is possible only when each of all the diagonal minors separately is equal to zero.

Moreover, it is known that the diagonal minors \(M_{ii}, M_{jj}\) are connected with the minors of the other terms of the same determinant, if it vanishes, by the relations:

\[ \frac{M_{i1}}{M_{j1}} = \frac{M_{i2}}{M_{j2}} = \frac{M_{i3}}{M_{j3}} = \ldots = \frac{M_{ii}}{M_{ij}} = \frac{M_{ij}}{M_{jj}} = \ldots = \frac{M_{ik}}{M_{jk}}, \]

from which it follows that \(M_{jj}\cdot M_{ii}=M_{ji}^{2}\) when \(s\beta\) is substituted; from this last equality it is seen that, simultaneously with the diagonal minors \(M_{ii}\) and \(M_{jj}\) at \(s=\beta\), the minor \(M_{ji}\) also vanishes. Running through all \(i\) and \(j\), we shall prove that all first minors vanish at \(s=\beta\).

From the very same expansion of the derivative of the determinant \(\Delta'(s)\) with respect to \(s\), i.e. from formula \((d)\), we shall draw the general conclusion that any \(\mu\)-fold root \(s=\gamma\) of the determinant \(\Delta'(s)\) will be a root of multiplicity \((\mu-1)\) for the derivative \(\dfrac{d\Delta'(s)}{ds}\), and consequently for each separate partial derivative \(\dfrac{\partial\Delta'(s)}{\partial U_{ii}}\), i.e. for all the diagonal minors of the determinant \(\Delta'(s)\); but by virtue of the formula

\[ M_{ii} M_{jj}=M_{ij}^{2}, \]

which holds for \(s=\gamma\), the root \(\gamma\) will be a \(2(\mu-1)\)-fold root of \(M_{ij}^{2}\).

Thus a root of any multiplicity \(\mu\) of the determinant \(\Delta'(s)\) is a root of all the first minors of this determinant, and its multiplicity will be \(\mu-1\).

Observing that every diagonal minor occupies exactly the same position with respect to all the second minors formed from it as the principal determinant occupies with respect to it, we may already conclude, without proof, that a \(\mu\)-fold root of the principal determinant will be a root of multiplicity \((\mu-1)\) for its first minors, a root of multiplicity \((\mu-2)\) for the second minors, and so on, and finally a simple root for the \((\mu-1)\)-st minors of the determinant \(\Delta'(s)\).

The system of harmonizing multipliers corresponding to some multiple root of \(\Delta'(s)\).

Since the derivation of differential equations of harmonic type presupposes the preliminary determination of the harmonizing multipliers, we shall solve this problem separately, assuming at first, for simplicity of reasoning, that the root of the equation \(\Delta'(s)=0\) is only double. In the preceding paragraph it was proved that a double root of the determinant \(\Delta'(s)=0\) is a root of all the first minors of this determinant; therefore we must abandon the idea of finding the harmonizing multipliers \(\alpha\) by formulas \((\alpha)\), and instead seek at least one second minor of the same determinant which would be different from zero upon substituting into it the double root of the original determinant.

Suppose that such a second minor, different from zero upon substitution of the double root, is the minor

\[ \frac{\partial^{2}\Delta'(s)} {\partial U'_{k-1\,k-1}\,\partial U_{kk}} = M\binom{k-1,\ k}{k-1,\ k}, \]

we shall write the system of linear equations determining the harmonizing multipliers \(\alpha\) for the double root \(s=\beta\).

Separating in this system the first \((k-2)\) equations, and transferring the last two terms of each of these equations from the left-hand side to the right-hand side, we shall have the following system of linear equations with respect to \(\alpha_i\) \((i=1,2,3\ldots k-2)\), with right-hand sides:

\[ \left\{ \begin{aligned} &(a_{11}\beta-b_{11})\alpha_1+(a_{12}\beta-b_{12})\alpha_2+\ldots +(a_{1\,k-2}\beta-b_{1\,k-2})\alpha_{k-2} \\ &\qquad =-(a_{1\,k-1}\beta-b_{1\,k-1})\alpha_{k-1} -(a_{1k}\beta-b_{1k})\alpha_k,\\[4pt] &(a_{21}\beta-b_{21})\alpha_1+(a_{22}\beta-b_{22})\alpha_2+\ldots +(a_{2\,k-2}\beta-b_{2\,k-2})\alpha_{k-2} \\ &\qquad =-(a_{2\,k-1}\beta-b_{2\,k-1})\alpha_{k-1} -(a_{2k}\beta-b_{2k})\alpha_k,\\[4pt] &(a_{31}\beta-b_{31})\alpha_1+(a_{32}\beta-b_{32})\alpha_2+\ldots +(a_{3\,k-2}\beta-b_{3\,k-2})\alpha_{k-2} \\ &\qquad =-(a_{3\,k-1}\beta-b_{3\,k-1})\alpha_{k-1} -(a_{3k}\beta-b_{3k})\alpha_k,\\ &\qquad \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\\ &\qquad \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\\ &(a_{k-2\,1}\beta-b_{k-2\,1})\alpha_1 +(a_{k-2\,2}\beta-b_{k-2\,2})\alpha_2+\ldots +(a_{k-2\,k-2}\beta-b_{k-2\,k-2})\alpha_{k-2} \\ &\qquad =-(a_{k-2\,k-1}\beta-b_{k-2\,k-1})\alpha_{k-1} -(a_{k-2\,k}\beta-b_{k-2\,k})\alpha_k . \end{aligned} \right. \tag{s} \]

The system \((s)\) proves to be solvable with respect to the multipliers \(\alpha_1\alpha_2\ldots \alpha_{k-2}\), since its determinant, or the second minor of the initial determinant, \(M\binom{k-1,k}{k-1,k}\), by our assumption is different from zero.

Solving these simple equations by the general rules, we obtain, for some \(\alpha_i\), the following expression:

\[ \alpha_i= -\frac{1}{M\binom{k-1,k}{k-1,k}} \left\{ \alpha_{k-1}\left|U_{11},U_{22},\ldots U_{i\,k-1},\ldots U_{k-2\,k-2}\right| +\alpha_k\left|U_{11},U_{22},\ldots U_{ik},\ldots U_{k-2\,k-2}\right| \right\}_{s=\beta}, \tag{\(\alpha_i\)} \]

where the equality \(s=\beta\) has been appended to the brace to denote the substitution everywhere, instead of \(s\), of the double root \(\beta\).

The last formula, as \(i\) varies from \(1\) to \(k-2\), gives the values of the first \(k-2\) multipliers \(\alpha\) as functions of the last two, \(\alpha_{k-1}\) and \(\alpha_k\), which remain completely arbitrary.

As for the last two equations of the system serving to determine the multipliers \(\alpha\), which we have not written down, these equations turn into identities upon substitution of the first \((k-2)\) multipliers

\[ \alpha_i=\alpha_{k-1} \frac{M\binom{n\ k-1}{n\ i}}{M\binom{n\ k-1}{n\ k-1}} +\alpha_k \frac{M\binom{n\ k-1}{n-1\ i}}{M\binom{n\ k-1}{n\ k-1}}, \]

and that, upon substituting these values of \(\alpha_i\) into the equations

\[ \left. \begin{aligned} &(a_{k-1,1}\beta-b_{k-1,1})\alpha_1 +(a_{k-1,2}\beta-b_{k-1,2})\alpha_2+\cdots +(a_{k-1,k-2}\beta-b_{k-1,k-2})\alpha_{k-2}\\ &\qquad +(a_{k-1,k-1}\beta-b_{k-1,k-1})\alpha_{k-1} +(a_{k-1,k}\beta-b_{k-1,k})\alpha_k=0,\\[4pt] &(a_{k1}\beta-b_{k1})\alpha_1 +(a_{k2}\beta-b_{k2})\alpha_2+\cdots +(a_{k,k-2}\beta-b_{k,k-2})\alpha_{k-2}\\ &\qquad +(a_{k,k-1}\beta-b_{k,k-1})\alpha_{k-1} +(a_{kk}\beta-b_{kk})\alpha_k=0 \end{aligned} \right\} \tag{G} \]

and reducing to a common denominator, the left-hand sides of these equations take the form of the following binomials:

\[ \frac{\alpha_{k-1}}{M\binom{n\ k-1}{n\ k-1}} \sum_{i=1}^{i=k-1} M\binom{n\ k-1}{n\ i}\,(a_{k-1,i}\beta-b_{k-1,i}) + \frac{\alpha_k}{M\binom{n\ k-1}{n\ k-1}} \sum_{i=1}^{i=k-1} M\binom{n\ k-1}{n-1\ i}\,(a_{k-1,i}\beta-b_{k-1,i}) \]

\[ \frac{\alpha_{k-1}}{M\binom{n\ k-1}{n\ k-1}} \sum_{i=1}^{i=k-2} M\binom{n\ k-1}{n\ i}\,(a_{ki}\beta-b_{ki}) + \frac{\alpha_k}{M\binom{n\ k-1}{n\ k-1}} \sum_{i=1}^{i=k-2} M\binom{n\ k-1}{n-1\ i}\,(a_{ki}\beta-b_{ki}) \]

\[ + \frac{ M\binom{n\ k-1}{n\ k-1}(a_{k,k-1}\beta-b_{k,k-1})\alpha_{k-1} + M\binom{n\ k-1}{n\ k-1}(a_{kk}\beta-b_{kk})\alpha_k }{ M\binom{n\ k-1}{n\ k-1} }; \]

the coefficients of \(\alpha_{k-1}\) and \(\alpha_k\) in these expressions are the expansions, with respect to the elements of the \((k-1)\)-st and \(k\)-th rows, of the minors \(M\binom{n}{n}\) and \(M\binom{n}{n-1}\) of the determinant \(\Delta(s)_{s=\beta}\).

But all the first minors, by what has been proved, are equal to zero; therefore the last two equations are satisfied identically by themselves for arbitrary values of the multipliers \(\alpha_{k-1}\) and \(\alpha_k\).

After the detailed exposition of the method for finding a system of harmonizing multipliers corresponding to a double root of the determinant \(\Delta'(s)\), the solution of the same problem for the case of a triple root is clear in itself. Since a triple root of the determinant \(\Delta'(s)\) is at the same time a root of all its second minors, then, for determining the multipliers \(\alpha_i\), one should, from the \(k\) equations to which

must satisfy these multipliers, one must choose only \(k-3\) equations. The choice of these equations must be made so that, with their help, \(k-3\) multipliers \(\alpha\) can be determined as functions of the other three, which themselves remain entirely arbitrary. Then, just as we did above, in the present case too we can make certain that the system of multipliers found, among which three are entirely arbitrary, will satisfy the last three equations of the system, which remained unused in finding the multipliers \(\alpha\).

Applying, with the appropriate modifications, the same method of determining the multipliers \(\alpha\) to the case when the discriminant \(\Delta'(s)\) has a \(\mu\)-fold root, we shall find that in this case, of the \(k\) multipliers \(\alpha\), \(\mu\) multipliers remain entirely arbitrary, and only \(k-\mu\) multipliers will be completely determined linear functions of the first \(\mu\).

Equations of harmonic type corresponding to multiple roots of the discriminant \(\Delta(s)\).

The peculiarities of the solution caused by the existence of a multiple root of the determinant \(\Delta'(s)\) may be observed in the simplest of cases, when the determinant just mentioned has one double root.

As has just been indicated, the harmonizing multipliers corresponding to the double root \(s=\beta\) have the form:

\[ \alpha_i=f_{k-1}^{(i)}\alpha_{k-1}+f_k^{(i)}\alpha_k \quad (i=1,2,3,\ldots k-2), \]

where, for brevity, it is set that

\[ f_{k-1}^{(i)}=\frac{\mathrm{M}\binom{k-1\ i}{i\ k}} {\mathrm{M}\binom{k\ k-1}{k\ k-1}} \quad \text{and} \quad f_k^{(i)}=\frac{\mathrm{M}\binom{k\ \ k-1}{k-1\ i}} {\mathrm{M}\binom{k\ k-1}{k\ k-1}}. \]

With their help the sum

\[ \sum_{i=1}^{i=k}\alpha_i Q_i=U_\beta \]

acquires the following form:

\[ U_\beta = \alpha_{k-1} \left( \sum_{i=1}^{i=k-2} f_{k-1}^{(i)}Q_i+Q_{k-1} \right) + \alpha_k \left( \sum_{i=1}^{i=k-2} f_k^{(i)}Q_i+Q_k \right) \]

or

\[ U_\beta=\alpha_{k-1}U_\beta^{(k-1)}+\alpha_k U_\beta^{(k)} , \]

where, for brevity, it has been put:

\[ \mathrm{U}_{\beta}^{(k)}=\sum_{i=1}^{i=k-2} f_{\kappa}^{(i)} Q_i+Q_{\kappa} \quad\text{and}\quad \mathrm{U}_{\beta}^{(\kappa-1)}=\sum_{i=1}^{i=\kappa-2} f_{\kappa-1}^{(i)} Q_i+Q_{\kappa-1}. \]

The differential equation

\[ \frac{d^2 \mathrm{U}_{\beta}}{dt^2}+\beta \mathrm{U}_{\beta}=0, \]

after factoring out the multipliers \(\alpha_{\kappa-1}\) and \(\alpha_{\kappa}\), is reduced to the form:

\[ \alpha_{\kappa-1}\left\{\frac{d^2 \mathrm{U}_{\beta}^{(\kappa-1)}}{dt^2} +\beta \mathrm{U}_{\beta}^{(\kappa-1)}\right\} +\alpha_{\kappa}\left\{\frac{d^2 \mathrm{U}_{\beta}^{(\kappa)}}{dt^2} +\beta \mathrm{U}_{\beta}^{(\kappa)}\right\}=0; \]

and, in view of the complete arbitrariness of these multipliers, such an equation is equivalent to two separate equations:

\[ \frac{d^2 \mathrm{U}_{\beta}^{(\kappa-1)}}{dt^2} +\beta \mathrm{U}_{\beta}^{(\kappa-1)}=0 \quad\text{and}\quad \frac{d^2 \mathrm{U}_{\beta}^{(\kappa)}}{dt^2} +\beta \mathrm{U}_{\beta}^{(\kappa)}=0. \]

From the manner in which the expressions \(\mathrm{U}_{\beta}^{(\kappa-1)}\) and \(\mathrm{U}_{\beta}^{(\kappa)}\) are formed, it is clear that both of them are the same kind of linear and homogeneous functions of the independent coordinates as are the functions \(U_h\), considered by us above, corresponding to simple roots of the discriminant \(\Delta'(s)\).

Extending the point of view developed by us on page (28) with respect to the polynomials \(U_h\) also to the polynomials \(\mathrm{U}_{\beta}^{(\kappa-1)}\) and \(\mathrm{U}_{\beta}^{(\kappa)}\), we shall come to the conclusion that, when there exists a double root of the discriminant \(\Delta(s)\), the two quadratic homogeneous functions \(2Q\) and \((2V-2V_0)\) can each be composed of the same \(k\) linear functions of the independent coordinates \(q_i\).

If, among the roots of the discriminant \(\Delta'(s)\), there were not a double root, but some \(\mu\)-fold root \(s=\lambda\), then, on the basis of the considerations set forth in the preceding paragraph, the harmonizing multipliers \(\alpha\) corresponding to this case would have the form:

\[ \alpha_i=f_{\kappa-\mu+1}^{(i)}\alpha_{\kappa-\mu+1} +f_{\kappa-\mu+2}^{(i)}\alpha_{\kappa-\mu+2} +\cdots+f_{\kappa-1}^{(i)}\alpha_{\kappa-1} +f_{\kappa}^{(i)}\alpha_{\kappa}, \]

where \(i=1,2,3,\ldots,k-\mu\), and where all \(f_{\kappa-e}^{(i)}\) \((e=0,1,2,3,\ldots,\mu-1)\) are functions of \(\lambda\), analogous in composition to \(f_{\kappa-1}^{(i)}\) and \(f_{\kappa}^{(i)}\).

The polynomial \(U_{\lambda}\), formed with the aid of the multipliers \(\alpha\) from \(Q_1,Q_2,\ldots,Q_{\kappa}\), will contain, as is not difficult to see, \(\mu\) pro-

arbitrary constants \(a_{\kappa-\ell}\) \((\ell=0,1,2,\ldots,\mu-1)\), and will have the form:

\[ \begin{aligned} U_\lambda={}&a_{\kappa-\mu+1}\bigl(f_{\kappa-\mu+1}^{(1)}Q_1+f_{\kappa-\mu+1}^{(2)}Q_2+\cdots+f_{\kappa-\mu+1}^{(\kappa-\mu)}Q_{\kappa-\mu}+Q_{\kappa-\mu+1}\bigr)\\ &+a_{\kappa-\mu+2}\bigl(f_{\kappa-\mu+2}^{(1)}Q_1+f_{\kappa-\mu+2}^{(2)}Q_2+\cdots+f_{\kappa-\mu+2}^{(\kappa-\mu)}Q_{\kappa-\mu}+Q_{\kappa-\mu+2}\bigr)\\ &+\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\\ &+a_{\kappa-1}\bigl(f_{\kappa-1}^{(1)}Q_1+f_{\kappa-1}^{(2)}Q_2+\cdots+f_{\kappa-1}^{(\kappa-\mu)}Q_{\kappa-\mu}+Q_{\kappa-1}\bigr)\\ &+a_{\kappa}\bigl(f_{\kappa}^{(1)}Q_1+f_{\kappa}^{(2)}Q_2+\cdots+f_{\kappa}^{(\kappa-\mu)}Q_{\kappa-\mu}+Q_{\kappa}\bigr) \end{aligned} \]

or

\[ \begin{aligned} U_\lambda={}&a_{\kappa-\mu+1} \left[ \sum_{r=1}^{r=\kappa-\mu} f_{\kappa-\mu+1}^{(r)}Q_r+Q_{\kappa-\mu+1} \right]\\ &+a_{\kappa-\mu+2} \left[ \sum_{r=1}^{r=\kappa-\mu} f_{\kappa-\mu+2}^{(r)}Q_r+Q_{\kappa-\mu+2} \right]\\ &+\cdots+a_{\kappa-1} \left[ \sum_{r=1}^{r=\kappa-\mu} f_{\kappa-1}^{(r)}Q_r+Q_{\kappa-1} \right] +a_{\kappa} \left[ \sum_{r=1}^{r=\kappa-\mu} f_{\kappa}^{(r)}Q_r+Q_{\kappa} \right]. \end{aligned} \]

Introducing for the polynomials enclosed in square brackets abbreviated notations by means of the letter \(U\) with two indices, of which one indicates the root \(\lambda\), the other the arbitrary multiplier \(a\) with the corresponding subscript, we shall write the polynomial \(U_\lambda\) in the form:

\[ U_\lambda = a_{\kappa-\mu+1}U_\lambda^{(\kappa-\mu+1)} +a_{\kappa-\mu+2}U_\lambda^{(\kappa-\mu+2)} +\cdots +a_{\kappa-1}U_\lambda^{(\kappa-1)} +a_{\kappa}U_\lambda^{(\kappa)}; \]

at the same time the polynomial \(V_\lambda\) will be equal to \(\lambda U_\lambda\), and the differential equation

\[ \frac{d^2 U_\lambda}{d t^2}+\lambda U_\lambda=0 \]

with these notations can be written in the following form:

\[ a_{\kappa-\mu+1} \left\{ \frac{d^2 U_\lambda^{(\kappa-\mu+1)}}{d t^2} +\lambda U_\lambda^{(\kappa-\mu+1)} \right\} +a_{\kappa-\mu+2} \left\{ \frac{d^2 U_\lambda^{(\kappa-\mu+2)}}{d t^2} +\lambda U_\lambda^{(\kappa-\mu+2)} \right\} \]

\[ +\cdots +a_{\kappa-1} \left\{ \frac{d^2 U_\lambda^{(\kappa-1)}}{d t^2} +\lambda U_\lambda^{(\kappa-1)} \right\} +a_{\kappa} \left\{ \frac{d^2 U_\lambda^{(\kappa)}}{d^2 t} +\lambda U_\lambda^{(\kappa)} \right\} =0. \]

In view of the arbitrariness of the multipliers $\alpha$, this equation splits into $\mu$ separate differential equations of the form:

\[ \frac{d^2 \mathrm{U}_{\lambda}^{(k-\mu+r)}}{d^2 t} +\lambda \mathrm{U}_{\lambda}^{(k-\mu+r)}=0 \quad (r=0,1,2,3,\ldots,\mu-1). \]

To the dependent variables of these differential equations
$\mathrm{U}_{\lambda}^{(k-\mu+r)}$ $(r=0,1,2,3,\ldots,\mu-1)$ all those observations apply which were made above with respect to the analogous expressions $\mathrm{U}_{\beta}^{(k-1)}$ and $\mathrm{U}_{\beta}^{(k)}$ (p. 44).

The number of such variables $\mathrm{U}_{\lambda}^{(k-\mu+r)}$ is always equal to the number expressing the multiplicity of the root $\lambda$, and all these variables are linearly independent of one another, since, for entirely arbitrary multipliers
$\alpha_{k-\mu+1}, \alpha_{k-\mu+2}, \ldots, \alpha_{k-1}, \alpha_k$, the sum

\[ \sum_{e=1}^{e=\mu} \alpha_{k-\mu+e}\, \mathrm{U}_{\lambda}^{(k-\mu+e)} = \mathrm{U}_{\lambda} \]

does not vanish.

Finding polynomials of the form $\mathrm{U}_{\lambda}^{(k-\mu+r)}$ by means of the transformation of the quadratic forms $2Q$ and $2V-2V_0$.

Since, by the preceding paragraph, to each multiple root $\lambda$ of the discriminant $\Delta(s)$ there corresponds a system of mutually independent polynomials $\mathrm{U}_{\lambda}^{(k-\mu+r)}$ in a number equal to the exponent of multiplicity of the root $\lambda$ under consideration, it is easy to arrive at the conclusion that, whatever the roots of the equation $\Delta(s)=0$ and their multiplicities may be, it is always possible to find $k$ distinct polynomials of the form $\mathrm{U}_h$ and $\mathrm{U}_{\lambda}^{(k-\mu+r)}$, and from the squares of these polynomials to compose both quadratic functions $2Q$ and $2V-2V_0$.

We shall confirm this conclusion, obtained as a consequence of our method of combining equations, by a direct transformation of the quadratic forms, by means of which the polynomials $\mathrm{U}_{\lambda}^{(k-\mu+r)}$ will at once be determined.

Not wishing to lengthen the formulae without need, we may suppose that the discriminant $\Delta(s)$, besides the simple roots $s=s_i$ $(i=1,2,3,\ldots,i)$, has also the roots $\lambda$ and $\lambda_1$ of multiplicities $\mu$ and $\mu_1$, and carry out the transformation for this case, since the presence of a larger number of multiple roots introduces no essential changes

in the process of transformation. In the transformation under consideration, we shall take as the initial expressions for \(2Q\) and \(2V-2V_0\) those obtained with the aid of Euler’s theorem, which have the form:

\[ 2Q=\sum_{e=1}^{e=k} Q_e q_e \quad \text{and} \quad 2V-2V_0=\sum_{e=1}^{e=k} V_e q_e, \]

where \(q_e\), according to the formula \((q_e)\) on page (35), is equal to

\[ \sum_{j=1}^{j=k} \frac{M_{ej}}{\Delta(s)} \Phi_j . \]

Before proceeding to further transformations of the quadratic forms, we note that, under our assumptions concerning the composition of the determinant \(\Delta(s)\), the fraction \(\frac{M_{ej}}{\Delta(s)}\), entering into the expression for \(q_e\), admits cancellation by the product \((s-\lambda)^{\mu-1}\cdot(s-\lambda_1)^{\mu_1-1}\), since the multiple roots of the determinant \(\Delta(s)\) are multiple roots of all its first minors, and their multiplicity in these minors is diminished by one.

Denoting

\[ \frac{M_{ej}}{(s-\lambda)^{\mu-1}\cdot(s-\lambda_1)^{\mu_1-1}} \]

by \(N_{je}\),

\[ \frac{\Delta(s)}{(s-\lambda)^{\mu-1}\cdot(s-\lambda_1)^{\mu_1-1}} \]

by \(\delta(s)\), we obtain for \(q_e\) the expression:

\[ q_e=\sum_{j=1}^{j=k} \frac{N_{ej}}{\delta(s)} \Phi_j . \]

In view of the complete independence of the variables \(q_e\) from the arbitrary factor \(s\), \(q_e\) can be set equal only to that term of the sum

\[ \sum_{j=1}^{j=k} \frac{N_{ej}}{\delta(s)} \Phi_j, \]

which is free of the factor \(s\), i.e. contains \(s\) in degree zero; the coefficients at all other powers of \(s\), except the zero—

left, must vanish. This requirement leads us to an infinite sequence of equalities, of which we shall write here only the first two:

\[ q_e=\sum_{j=1}^{j=\kappa}\left[\frac{\mathrm N_{ej}}{\delta(s)}\Phi_j\right]_{s^0} \quad\text{and}\quad \sum_{j=1}^{j=\kappa}\left[\frac{\mathrm N_{ej}}{\delta(s)}\Phi_j\right]_{s^{-1}}=0 \tag{s} \]

where the brackets with the subscripts \(s^0\) and \(s^{-1}\) denote symbolically the coefficients of \(s^0\) and \(s^{-1}\) in the expansion of the fraction \(\dfrac{\mathrm N_{ej}\Phi_j}{\delta(s)}\) in descending powers of \(s\).

Since the degree of \(\mathrm N_{ej}\) with respect to \(s\) is only one less than the degree of \(\delta(s)\), and \(\Phi_j=sQ_j-V_j\), it follows that in the expansion the highest terms will be the terms of zero degree in \(s\), obtained by dividing the highest terms of the products \(\mathrm N_{ej}\Phi_j\), i.e. \(s\mathrm N_{ej}Q_j\), by \(\delta(s)\); in other words, \(q_e\) can be expressed thus:

\[ q_e= \sum_{j=1}^{j=\kappa}\left[\frac{\mathrm N_{ej}\Phi_j}{\delta(s)}\right]_{s^0} = \sum_{j=1}^{j=\kappa}\left[\frac{s\mathrm N_{ej}Q_j}{\delta(s)}\right]_{s^0} = \sum_{j=1}^{j=\kappa}\left[\frac{\mathrm N_{ej}Q_j}{\delta(s)}\right]_{s^{-1}} = \sum_{j=1}^{j=\kappa}\left[\frac{\mathrm N_{ej}}{\delta(s)}\right]_{s^{-1}}Q_j. \]

In exactly the same way the second of formulas \((s)\) gives:

\[ \sum_{j=1}^{j=\kappa}\left[\frac{\mathrm N_{ej}\Phi_j}{\delta(s)}\right]_{s^{-1}} = \sum_{j=1}^{j=\kappa}\left[\frac{s\mathrm N_{ej}Q_j}{\delta(s)}\right]_{s^{-1}} - \sum_{j=1}^{j=\kappa}\left[\frac{\mathrm N_{ej}V_j}{\delta(s)}\right]_{s^{-1}} =0, \]

or

\[ \sum_{j=1}^{j=\kappa}\left[\frac{s\mathrm N_{ej}Q_j}{\delta(s)}\right]_{s^{-1}} = \sum_{j=1}^{j=\kappa}\left[\frac{\mathrm N_{ej}V_j}{\delta(s)}\right]_{s^{-1}}; \]

but

\[ \sum_{j=1}^{j=\kappa}\left[\frac{s\mathrm N_{ej}Q_j}{\delta(s)}\right]_{s^{-1}} = \sum_{j=1}^{j=\kappa}\left[\frac{\mathrm N_{ej}Q_j}{\delta(s)}\right]_{s^{-2}}, \]

and therefore

\[ \sum_{j=1}^{j=\kappa}\left[\frac{\mathrm N_{ej}V_j}{\delta(s)}\right]_{s^{-1}} = \sum_{j=1}^{j=\kappa}\left[\frac{\mathrm N_{ej}Q_j}{\delta(s)}\right]_{s^{-2}}, \]

or

\[ \sum_{j=1}^{j=\kappa}\left[\frac{\mathrm N_{ej}}{\delta(s)}\right]_{s^{-1}}V_j = \sum_{j=1}^{j=\kappa}\left[\frac{\mathrm N_{ej}}{\delta(s)}\right]_{s^{-2}}Q_j. \]

Using the formulas derived, we shall give the quadratic forms \(2Q\) and \(2V-2V_0\) the following form:

\[ 2Q=\sum_{e=1}^{e=\kappa} Q_e q_e = \sum_{e=1}^{e=\kappa}\sum_{j=1}^{j=\kappa} \left[\frac{N_{ej}}{\delta(s)}\right]_{s-1} Q_e Q_j \]

\[ \begin{aligned} 2V-2V_0 &=\sum_{e=1}^{e=\kappa} V_e q_e = \sum_{e=1}^{e=\kappa}\sum_{j=1}^{j=\kappa} \left[\frac{N_{ej}}{\delta(s)}\right]_{s-1} Q_j V_e \\ &= \sum_{j=1}^{j=\kappa}\sum_{e=1}^{e=\kappa} \left[\frac{N_{ej}}{\delta(s)}\right]_{s-1} V_e Q_j \\ &= \sum_{e=1}^{e=\kappa}\sum_{j=1}^{j=\kappa} \left[\frac{N_{ej}}{\delta(s)}\right]_{s-2} Q_e Q_j . \end{aligned} \]

The coefficients of the reduced quadratic forms

\[ \left[\frac{N_{ej}}{\delta(s)}\right]_{s-1} \quad \text{and} \quad \left[\frac{N_{ej}}{\delta(s)}\right]_{s-2} \]

can be computed on the basis of the following considerations: since
\(\dfrac{N_{ej}(s)}{\delta(s)}\) is a proper fraction whose denominator \(\delta(s)\) has only simple roots
\(s_1, s_2, s_3,\ldots, s_i, \lambda\), and \(\lambda_1\), then

\[ \frac{N_{je}(s)}{\delta(s)} = \sum_{i=1}^{i=i}\frac{A_{ej}^{(s_i)}}{s-s_i} + \frac{A_{ej}^{(\lambda)}}{s-\lambda} + \frac{A_{ej}^{(\lambda_1)}}{s-\lambda_1} = \sum_{i=1}^{i=i+2}\frac{A_{ej}^{(i)}}{s-s_i}, \]

where \(\lambda\) and \(\lambda_1\) are included in the general count of the roots \(s_i\).

Taking the factor \(s\) outside the denominator of the partial fraction
\(\dfrac{A_{ej}^{(i)}}{s-s_i}\), we shall have:

\[ \frac{A_{ej}^{(i)}}{s-s_i} = \frac{A_{ej}^{(i)}}{s\left(1-\frac{s_i}{s}\right)} = A_{ej}^{(i)} s^{-1} \left(1-\frac{s_i}{s}\right)^{-1} = A_{ej}^{(i)} s^{-1} \left\{1+\frac{s_i}{s}+\frac{s_i^2}{s^2}+\cdots\right\}, \]

or

\[ \frac{A_{ej}^{(i)}}{s-s_i} = A_{ej}^{(i)} s^{-1} + A_{ej}^{(i)} s_i s^{-2} + A_{ej}^{(i)} s_i^2 s^{-3} +\cdots ; \]

summing the partial fractions with respect to the index \(i\) from \(i=1\) to \(i=i+2\), we find:

\[ \sum_{i=1}^{i=i+2}\frac{A_{ej}^{(i)}}{s-s_i} = s^{-1}\sum_{i=1}^{i=i+2} A_{ej}^{(i)} + s^{-2}\sum_{i=1}^{i=i+2} A_{ej}^{(i)} s_i +\cdots . \]

From this expansion it is clear that

\[ \left[\frac{N_{ej}(s)}{\delta(s)}\right]_{s^{-1}} = \sum_{i=1}^{i=i+2} A_{ej}^{(i)} \quad\text{and}\quad \left[\frac{N_{ej}(s)}{\delta(s)}\right]_{s^{-2}} = \sum_{i=1}^{i=i+2} A_{ej}^{(i)} s_i . \tag{k} \]

Using the obtained values of the coefficients

\[ \left[\frac{N_{ej}}{\delta(s)}\right]_{s^{-1}} \quad\text{and}\quad \left[\frac{N_{ej}}{\delta(s)}\right]_{s^{-2}}, \]

we shall write the quadratic forms \(2Q\) and \(2V-2V_0\) in the following form:

\[ 2Q = \sum_{e=1}^{e=\kappa} \sum_{j=1}^{j=\kappa} \sum_{i=1}^{i=i+2} A_{ej}^{(i)} Q_e Q_j , \]

\[ 2V-2V_0 = \sum_{e=1}^{e=\kappa} \sum_{j=1}^{j=\kappa} \sum_{i=1}^{i=i+2} s_i A_{ej}^{(i)} Q_e Q_j . \]

By moving the summation over \(i\) to the last place, we shall give these formulas the following form:

\[ 2Q = \sum_{i=1}^{i=i+2} \left\{ \sum_{e=1}^{e=\kappa} \sum_{j=1}^{j=\kappa} A_{ej}^{(i)} Q_e Q_j \right\} = \sum_{i=1}^{i=i+2} P_i , \]

\[ 2V-2V_0 = \sum_{i=1}^{i=i+2} s_i \left\{ \sum_{e=1}^{e=\kappa} \sum_{j=1}^{j=\kappa} A_{ej}^{(i)} Q_e Q_j \right\} = \sum_{i=1}^{i=i+2} s_i P_i , \]

where by \(P_i\) are denoted the quadratic forms

\[ P_i = \sum_{e=1}^{e=\kappa} \sum_{j=1}^{j=\kappa} A_{ej}^{(i)} Q_e Q_j, \]

in which the coefficients \(A_{je}^{(i)}\) may, if desired, be replaced by their values according to the known formulas

\[ A_{ej}^{(i)} = N_{ej}(s_i) : \left(\frac{d\delta}{ds}\right)_{s=s_i}. \]

Using the formula

\[ \delta(s) = \frac{\Delta(s)} {(s-\lambda)^{\mu-1}(s-\lambda_1)^{\mu_1-1}}, \]

relating \(\delta(s)\) to \(\Delta(s)\), we shall find, by logarithmic differentiation, the following relation between their derivatives:

\[ \frac{1}{\delta(s)}\cdot \frac{d\delta}{ds} = \frac{1}{\Delta(s)}\cdot \frac{d\Delta}{ds} - \frac{\mu-1}{s-\lambda} - \frac{\mu_1-1}{s-\lambda_1}, \]

or

\[ \begin{aligned} \frac{d\delta}{ds} &= \frac{\delta(s)}{\Delta(s)}\cdot \frac{d\Delta}{ds} - (\mu-1)\frac{\delta(s)}{s-\lambda} - (\mu_1-1)\frac{\delta(s)}{s-\lambda_1} \\ &= \frac{1}{(s-\lambda)^{\mu-1}(s-\lambda_1)^{\mu_1-1}} \cdot \frac{d\Delta(s)}{ds} - \{(\mu-1)(s-\lambda_1)+(\mu_1-1)(s-\lambda)\} \prod_{1}^{i}(s-s_i). \end{aligned} \tag{II} \]

From this relation between the derivatives, upon substituting for \(s\) one of the roots \(s_i\), we find:

\[ \left(\frac{d\delta}{ds}\right)_{s=s_i} = \left(\frac{d\Delta}{ds}\right)_{s=s_i} \cdot \frac{1}{(s_i-\lambda)^{\mu-1}(s_i-\lambda_1)^{\mu_1-1}}, \tag{D} \]

since the second term of the right-hand side in formula (II), for every simple root \(s_i\) \((i=1,2,3,\ldots,i)\), vanishes identically. If we use this last relation between

\[ \left(\frac{d\delta}{ds}\right)_{s=s_i} \quad \text{and} \quad \left(\frac{d\Delta}{ds}\right)_{s=s_i} \]

to calculate the coefficients \(A_{ej}^{(i)}\), then we obtain

\[ A_{ej}^{(i)} = \frac{ N_{ej}(s_i)(s_i-\lambda)^{\mu-1}(s_i-\lambda_1)^{\mu_1-1} }{ \left(\dfrac{d\Delta}{ds}\right)_{s=s_i} } = \frac{ M_{ej}(s_i) }{ \left(\dfrac{d\Delta}{ds}\right)_{s=s_i} }. \]

This equality, which holds for all \(i\) from \(1\) to \(i\), shows that the coefficients \(A_{ej}^{(i)}\), in their structure, are identical with the corresponding coefficients found by us above for the case when the determinant \(\Delta(s)\) had only simple roots \(s_i\). Therefore all the quadratic forms \(P_i\) \((i=1,2,3,\ldots,i)\) will, in structure, be identical with the expressions \(\omega_i^2\) found above.

But for \(i=i+1\) and \(i=i+2\) formula (D) will not hold, since

\[ \prod_{1}^{i}(s-s_i) \]

for \(s=\lambda\) and \(s=\lambda_1\) in formula (II) does not vanish; thus the polynomials \(P_{i+1}\) and \(P_{i+2}\), corresponding to the multiple roots \(\lambda\) and \(\lambda_1\) of the determinant \(\Delta(s)\), while remaining quadratic and homogeneous functions of \(Q_i\), and consequently also of \(q_i\), in their

the structure nevertheless deviate somewhat from the general type of the polynomials \(P_i\) or \(\omega_i^2\). This difference of the polynomials \(P_{i+1}\) and \(P_{i+2}\) from all the other polynomials \(P_i\) \((i=1,2,3,\ldots i)\) consists in the fact that the first two polynomials are not exact squares, but can only be decomposed into squares of linear homogeneous expressions. The complete analogy between the expressions \(P_{i+1}\) and \(P_{i+2}\) allows us to confine ourselves to transforming only one of these expressions.

Choosing the first of these expressions

\[ P_{i+1}=\sum_{e=1}^{e=k}\sum_{j=1}^{j=k} A_{ej}^{(i+1)} Q_e Q_j, \]

we shall first establish one property of the coefficients of this expression \(A_{ej}^{(i+1)}\), namely that the determinant of this form and all its minors from the first to the \(k-\mu-1\)-st order, which will here be denoted respectively by the symbols

\[ \left|A_{ej}^{(i+1)}\right|_k,\quad \left|A_{ej}^{(i+1)}\right|_{k-1},\ldots,\left|A_{ej}^{(i+1)}\right|_{\mu+1}, \]

vanish identically. On the basis of the relation between minors whose orders differ by one, the proof of this property may be reduced to proving the vanishing of only the minors of order \(k-\mu-1\), since the vanishing of all these minors entails the vanishing of all the other minors and, finally, of the determinant \(\left|A_{ej}^{(i+1)}\right|_k\).

Wishing to prove the above-mentioned property with respect to one of the minors (for example, the diagonal one) of order \(k-\mu-1\), we shall regard this minor as the result of substituting the root \(s=\lambda\) into the determinant of order \(\mu+1\)

\[ \left|A_{ej}\right|_{\mu+1} = \left|A_{11} A_{22}\ldots A_{\mu+1\,\mu+1}\right|, \]

where

\[ A_{ej}=\frac{N_{ej}(s)}{\dfrac{d\delta}{ds}} = \frac{M_{ej}(s)} {\dfrac{d\delta}{ds}(s-\lambda)^{\mu-1}(s-\lambda_1)^{\mu_1-1}}. \]

After substituting these values in place of \(A_{ej}\), the determinant \(\left|A_{ej}\right|_{\mu+1}\) assumes the following form:

\[ \left|A_{ej}\right|_{\mu+1} = \left(\frac{d\delta}{ds}\right)^{-(\mu+1)} \cdot \left|N_{ej}\right|_{\mu+1} \]

\[ = \left(\frac{d\delta}{ds}\right)^{-(\mu+1)} \cdot (s-\lambda)^{-(\mu+1)(\mu-1)} \cdot (s-\lambda_1)^{-(\mu+1)(\mu_1-1)} \cdot \left|M_{ej}\right|_{\mu+1}. \]

But the determinant \(\left|M_{ej}\right|_{\mu+1}\), composed of the first minors of another determinant \(\Delta(s)\), is divisible by the \(\mu\)-th power of this latter, and therefore may be replaced by the product \(\Delta^\mu(s)\cdot F(s)\), which, after substituting, in place of \(\Delta(s)\), the product \(G(s)\cdot (s-\lambda)^\mu (s-\lambda_1)^{\mu_1}\), becomes

\[ \Delta^\mu \cdot F(s)=F(s)\cdot G^\mu(s)(s-\lambda)^{\mu^2}(s-\lambda_1)^{\mu\mu_1}. \]

Introducing the latter expression into the formula expressing \(\left|A_{ej}\right|_{\mu+1}\), and reducing, we find the final formula

\[ \left|A_{ej}\right|_{\mu+1} = F(s)\cdot G^{(\mu)}(s) \left(\frac{d\delta}{ds}\right)^{-(\mu+1)} \cdot (s-\lambda)\cdot (s-\lambda_1)^{\mu-\mu_1+1}, \]

from which it is seen that the substitution \(s=\lambda\) makes this determinant vanish. Applying the very same transformation procedure to any other minor of the same order, we would find that this minor too contains the factor \(s-\lambda\) and vanishes upon substituting the root \(s=\lambda\); but in a determinant of order \(\mu\) formed from these same coefficients \(A_{ej}\), the factor \(s-\lambda\) enters only to the zeroth degree; therefore the minors of order \(k-\mu\) are the first minors which do not vanish for \(s=\lambda\).

On the basis of the proved property of the coefficients \(A_{ej}^{(i+1)}\), which, in order to indicate their relation with the multiple root \(\lambda\), we shall denote by the symbol \(\lambda_{ej}\), the polynomial \(P_{i+1}\) may be transformed into the form of a sum of \(\mu\) squares.

We shall carry out this transformation in the shortest way if we perform the operation described by us above (on p. 19) on the determinant

\[ \Theta_\mu = \left| \begin{array}{cccccc} \lambda_{11}, & \lambda_{12}, & \lambda_{13}, & \ldots & \lambda_{1\mu}, & \pi_1 \\ \lambda_{21}, & \lambda_{22}, & \lambda_{23}, & \ldots & \lambda_{2\mu}, & \pi_2 \\ \lambda_{31}, & \lambda_{32}, & \lambda_{33}, & \ldots & \lambda_{3\mu}, & \pi_3 \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \lambda_{\mu 1}, & \lambda_{\mu 2}, & \lambda_{\mu 3}, & \ldots & \lambda_{\mu\mu}, & \pi_\mu \\ \pi_1, & \pi_2, & \pi_3, & \ldots & \pi_\mu, & P_{i+1} \end{array} \right|, \]

which is obtained by bordering the determinant

\[ \left|\lambda_{11},\,\lambda_{22},\,\lambda_{33},\ldots,\lambda_{\mu\mu}\right| = \left|A_{ej}^{(i+\mu)}\right|_{\mu} = |\lambda|_\mu \]

with the elements \(\pi_j\), where \(\pi_j\) is half the derivative

\[ \frac{1}{2}\frac{\partial P_{i+1}}{\partial Q_j} =\lambda_{j1}Q_1+\lambda_{j2}Q_2+\lambda_{j3}Q_3+\cdots+\lambda_{jk}Q_k . \]

Denoting the first minors of the determinant \(\Theta_\mu\) corresponding to its elements \(\lambda_{\mu\mu}\) and \(\pi_\mu\), respectively, by \(\Theta_{\mu-1}\) and \(H_\mu\), and the second minor of it, corresponding to the elements \(\lambda_{\mu\mu}\) and \(P_{i+1}\), i.e. \(\left|A^{(i+1)}_{\varrho j}\right|_{\mu-1}\), by \(|\lambda|_{\mu-1}\), we shall have the following formula:

\[ \Theta_\mu \cdot \begin{vmatrix} \Theta_{\mu-1}, & H_\mu\\ H_\mu, & |\lambda|_\mu \end{vmatrix} =\Theta_\mu^{\,2}\cdot|\lambda|_{\mu-1}, \]

or, after cancellation by \(\Theta_\mu\), we find:

\[ \begin{vmatrix} \Theta_{\mu-1}, & H_\mu\\ H_\mu, & |\lambda|_\mu \end{vmatrix} =\Theta_\mu\cdot|\lambda|_{\mu-1}. \]

Expanding the determinant standing on the left-hand side of this equality, we shall have the following formula:

\[ |\lambda|_\mu\cdot\Theta_{\mu-1}-H_\mu^{\,2} =\Theta_\mu\cdot|\lambda|_{\mu-1}, \]

to which we shall give a more symmetric form by dividing both sides of the equality by the product \(|\lambda|_\mu\cdot|\lambda|_{\mu-1}\); then the equation obtained is:

\[ (\Theta)\qquad \frac{\Theta_{\mu-1}}{|\lambda|_{\mu-1}} -\frac{H_\mu^{\,2}}{|\lambda|_\mu\cdot|\lambda|_{\mu-1}} = \frac{\Theta_\mu}{|\lambda|_\mu}, \quad\text{or}\quad \frac{\Theta_{\mu-1}}{|\lambda|_{\mu-1}} = \frac{\Theta_\mu}{|\lambda|_\mu} + \frac{H_\mu^{\,2}}{|\lambda|_\mu\cdot|\lambda|_{\mu-1}} . \]

Noting that, by virtue of the definitions,

\[ |\lambda|_1=\lambda_{11},\quad \Theta_0=P_{i+1},\quad H_1=\pi_1\quad\text{and}\quad \Theta_1=\lambda_{11}P_{i+1}-\pi_1^{\,2}, \]

and that, on the other hand, from formula \((\Theta)\) for \(\mu=1\)

\[ \Theta_1=\frac{\lambda_{11}}{|\lambda|_0}\cdot P_{i+1} -\frac{H_1^{\,2}}{|\lambda|_0}, \]

we find from the equation of the two expressions for \(\Theta_1\) that \(|\lambda|_0=1\).

Summing now the equalities \((\Theta)\) from \(\mu=1\) to \(\mu=\mu\), we find

\[ \sum_{\mu=1}^{\mu=\mu} \frac{\Theta_{\mu-1}}{|\lambda|_{\mu-1}} = \sum_{\mu=1}^{\mu=\mu} \frac{\Theta_\mu}{|\lambda|_\mu} + \sum_{\mu=1}^{\mu=\mu} \frac{H_\mu^{\,2}}{|\lambda|_\mu\cdot|\lambda|_{\mu-1}} . \]

After canceling the common terms in the sums

\[ \sum_{\rho=1}^{\rho=\mu}\frac{\Theta_{\mu-1}}{|\lambda|_{\mu-1}} \quad\text{and}\quad \sum_{\rho=1}^{\rho=\mu}\frac{\Theta_{\mu}}{|\lambda|_{\mu}} \]

we obtain:

\[ \frac{\Theta_0}{|\lambda|_0} = \frac{\Theta_\mu}{|\lambda|_\mu} + \sum_{\rho=1}^{\rho=\mu} \frac{H_\mu^2}{|\lambda|_\mu|\lambda|_{\mu-1}}, \]

or

\[ P_{i+1} = \frac{\Theta_\mu}{|\lambda|_\mu} + \sum_{\rho=1}^{\rho=\mu} \frac{H_\mu^2}{|\lambda|_\mu|\lambda|_{\mu-1}}. \]

But on the basis of the property of the coefficients \(\lambda_{ej}\) proved above, it is not difficult to prove that the determinant \(\Theta_\mu\) vanishes. Indeed, by expanding the elements of the last column, we shall obtain from the determinant \(\Theta_\mu\) \(k\) determinants \(\Theta_\mu^{(i)}\) \((i=1,2,3,\ldots k)\), differing from one another and from the determinant \(\Theta_\mu\) only in the elements of the last column. Any one of these determinants:

\[ \Theta_\mu^{(i)} = \left| \begin{array}{ccccc} \lambda_{11}, & \lambda_{12}, & \ldots & \lambda_{1\mu}, & \lambda_{1i}Q_i\\ \lambda_{21}, & \lambda_{22}, & \ldots & \lambda_{2\mu}, & \lambda_{2i}Q_i\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ \lambda_{\mu1}, & \lambda_{\mu2}, & \ldots & \lambda_{\mu\mu}, & \lambda_{\mu i}Q_i\\ \pi_1, & \pi_2, & \ldots & \pi_\mu, & \pi_iQ_i \end{array} \right| = \left| \begin{array}{ccccc} \lambda_{11}, & \lambda_{12}, & \ldots & \lambda_{1\mu}, & \lambda_{1i}\\ \lambda_{21}, & \lambda_{22}, & \ldots & \lambda_{2\mu}, & \lambda_{2i}\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ \lambda_{\mu1}, & \lambda_{\mu2}, & \ldots & \lambda_{\mu\mu}, & \lambda_{\mu i}\\ \pi_1, & \pi_2, & \ldots & \pi_\mu, & \pi_i \end{array} \right| \cdot Q_i, \]

for every \(i\), vanishes.

Indeed, so long as the index \(i\leq \mu\), the determinants standing as factors of \(Q_i\) vanish as determinants with two identical columns; but when the index \(i\) becomes greater than \(\mu\), the determinant standing as a factor of \(Q_i\) (after successively multiplying its horizontal rows by \(Q_1,Q_2,Q_3,\ldots Q_\mu\) and subtracting from the last) is transformed into the form:

\[ \left| \begin{array}{ccccc} \lambda_{11}, & \lambda_{12}, & \ldots & \lambda_{1\mu}, & \lambda_{1i}\\ \lambda_{21}, & \lambda_{22}, & \ldots & \lambda_{2\mu}, & \lambda_{2i}\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ \lambda_{\mu1}, & \lambda_{\mu2}, & \ldots & \lambda_{\mu\mu}, & \lambda_{\mu i}\\ \displaystyle\sum_{j=\mu+1}^{j=k}\lambda_{1j}Q_j, & \displaystyle\sum_{j=\mu+1}^{j=k}\lambda_{2j}Q_j, & \ldots & \displaystyle\sum_{j=\mu+1}^{j=k}\lambda_{\mu j}Q_j, & \displaystyle\sum_{j=\mu+1}^{j=k}\lambda_{ij}Q_j \end{array} \right|; \]

this same determinant, expanded with respect to the elements of the last row, is reduced to a sum of terms of the form:

\[ Q_j\cdot \begin{vmatrix} \lambda_{11} & \lambda_{12} & \ldots & \lambda_{1\mu} & \lambda_{1i} \\ \lambda_{21} & \lambda_{22} & \ldots & \lambda_{2\mu} & \lambda_{2i} \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \lambda_{\mu 1} & \lambda_{\mu 2} & \ldots & \lambda_{\mu\mu} & \lambda_{\mu i} \\ \lambda_{j1} & \lambda_{j2} & \ldots & \lambda_{j\mu} & \lambda_{ji} \end{vmatrix}, \]

in which the multipliers of \(Q_j\) are determinants of order \(\mu+1\) formed from the coefficients \(\lambda_{ji}\), and by the condition are equal to zero.

Setting all \(\Theta_\mu^{(i)}\), and consequently also \(\Theta_\mu\), equal to zero reduces the polynomial \(P_{i+1}\) to the form of a sum of \(\mu\) squares:

\[ P_{i+1}=\sum_{\mu=1}^{\mu=\mu}\frac{H_\mu^2}{|\lambda|_\mu\,|\lambda|_{\mu-1}}. \]

Repeating the same transformations as applied to the polynomial \(P_{i+2}\), corresponding to the multiple root \(\lambda_1\), we would find that this polynomial can be expanded into a sum of \(\mu_1\) squares of expressions linear with respect to \(Q_i\), and therefore also with respect to \(q_i\).

Since the quadratic forms \(2Q\) and \(2V-2V_0\) are both definite positive forms, all terms of the form

\[ \frac{H_\mu^2}{|\lambda|_\mu\,|\lambda|_{\mu-1}} \]

can only be positive quantities; in other words, the polynomials linear with respect to \(Q_i\) (or with respect to \(q_i\))

\[ \frac{H_\mu}{\sqrt{|\lambda|_\mu\,|\lambda|_{\mu-1}}} \]

will be polynomials with real coefficients. The same property, evidently, is also possessed by polynomials of the form

\[ \frac{H_\mu}{\sqrt{|\lambda_1|_\mu\,|\lambda_1|_{\mu-1}}}, \]

which correspond to another multiple root \(\lambda_1\).

It should also be noted here one more important property of the polynomials just mentioned, consisting in the fact that, both among the polynomials

\[ \frac{H_{\mu}}{\sqrt{|\lambda|_{\mu}\,|\lambda|_{\mu-1}}}, \]

belonging to one and the same multiple root, and among polynomials belonging to different (simple or multiple) roots, there is no linear dependence. To convince oneself of the existence of this property, it is enough to observe that the existence of a linear dependence among any two (or more) polynomials of the above-mentioned type would make it possible to combine into one term the squares of two linearly dependent polynomials entering into the quadratic forms, and thus to represent these forms as a sum of squares of polynomials linearly independent of one another, in a number smaller than the number \(k\).

But in that case, in order for the forms \(2Q\) and \(2V-2V_0\) to vanish, which must occur only for the values

\[ q_1=q_2=q_3=\cdots=q_k=0, \]

it would be sufficient to set equal to zero polynomials independent of one another; this would lead to a system of linear equations with the number of variables \(q_i\) greater than the number of equations, i.e. to establishing a functional dependence among the coordinates \(q_i\) independent of one another, the impossibility of which is directly evident.

From all that has been said regarding polynomials of the form

\[ \frac{H_{\mu}}{\sqrt{|\lambda|_{\mu}\,|\lambda|_{\mu-1}}}, \]

it is easy to conclude that these polynomials possess the very same properties as the polynomials defined above (p. 45)

\[ U_{\lambda}^{(k-\mu+r)}. \]

Integration of Lagrange’s equations when the discriminant \(\Delta(s)\) has both simple and multiple roots.

Keeping to the rather general assumptions made above (p. 46) concerning the roots of the discriminant \(\Delta(s)\), we shall obtain from the fundamental Lagrange equations three groups of differential equations. The first group consists of the equations connected with the simple roots of the discriminant \(\Delta(s)\); the second and third groups—the equations ...

equations associated with the multiple roots \(\lambda\) and \(\lambda_1\) of the same discriminant. The method of forming the equations of each of these groups has been described in detail above on p. 44; there it was also observed that the dependent variables of these equations \(U_h\) and \(U_{\lambda}^{(k-\mu+r)}\) are mutually independent linear functions of the parameters \(q_i\).

For convenience, one may introduce, for all such polynomials, a single general numbering and regard the symbol \(U_h\) as denoting the polynomials corresponding to the simple roots of the discriminant, so long as \(h=1,2,3\ldots i\); then the same symbol \(U_h\) denotes the polynomials corresponding to the multiple root \(\lambda\) for \(h=i+1,\ i+2,\ i+3,\ldots i+\mu\), and to the multiple root \(\lambda_1\) for \(h=i+\mu+1,\ i+\mu+2,\ i+\mu+3,\ i+\mu+\mu_1\).

Since, under this new convention, the total number of polynomials \(U_h\) is always equal to the number \(k\), i.e. to the number of independent parameters \(q_i\) of the system, then, by virtue of their mutual independence, they may be taken as new coordinates of the system, which are called the normal or principal coordinates of the system. The distinguishing feature of these coordinates is that they are integrals of differential equations of harmonic type,

\[ \frac{d^2 U_h}{d t^2}+s_h U_h=0, \]

and for this reason they are sometimes also called harmonic coordinates.

The general integral of this kind of differential equations has already been given by us above (formula \((k)\) on p. 27).

It is useful to add to what was said there a further remark. On the basis of the considerations set out on p. 25 and above, all roots \(s_h\) of the discriminant \(\Delta(s)\) or \(\Delta'(s)\) will be positive if the form \(2V-2V_0\) is a definite positive form, i.e. if the equilibrium position about which the system oscillates is a position of stable equilibrium.

In this case the right-hand side of equation \((k)\) can be transformed, by means of Euler’s formulas, into the form

\[ (k') \qquad U_h=E_h\sin(n_h t+\varepsilon_h), \]

where \(E_h\) and \(\varepsilon_h\) are arbitrary constants having a mechanical meaning; \(E_h\) is the so-called amplitude, and \(\varepsilon_h\) is the initial phase of the oscillation with period

\[ (T) \qquad T_h=\frac{2\pi}{n_h}=\frac{2\pi}{\sqrt{s_h}}. \]

To each simple root \(s_h\) \((h = 1, 2, 3, \ldots i)\) there corresponds one harmonic coordinate \(U_h\) with a definite period, amplitude, and initial phase; with each of the multiple roots \(\lambda\) and \(\lambda_1\) there are associated as many harmonic coordinates \(U_h\) as there are units contained in the exponents of their multiplicity.

All harmonic coordinates associated with one and the same multiple root \(\lambda\) (or \(\lambda_1\)) have one and the same period

\[ T_\lambda=\frac{2\pi}{\sqrt{\lambda}}\left(\text{or }T_{\lambda_1}=\frac{2\pi}{\sqrt{\lambda_1}}\right), \]

but their amplitudes and initial phases may vary from one coordinate to another. The reason for this phenomenon lies in the quite obvious circumstance that the periodic time (as is seen from formula (T)) depends on the roots of the discriminant \(\Delta(s)\), which in turn are functions of the coefficients of the quadratic forms \(2Q\) and \(2V-2V_0\), representing the kinetic and potential energy of the oscillating system. The variety of the initial circumstances of the motion, i.e. of the initial displacements and initial velocities of the various points of the system, while in no way affecting the periods of the harmonic coordinates, is reflected entirely only in their initial phases and amplitudes.

Expressions for the independent coordinates of the oscillating system and their derivatives.

The independent coordinates of the system \(q_i\) are found by the simple solution, with respect to these variables, of the system of equations \((k')\) on p. 58.

By virtue of the linear independence of the polynomials \(U_h\), such a solution is always possible and leads to the following expressions for \(q_i\):

\[ q_i=\sum_{h=1}^{h=k} C_h^{(i)} E_h \sin(n_h t+\varepsilon_h)\quad (i=1,2,3,\ldots k), \]

where \(C_h^{(i)}\) represents a fraction whose denominator is the determinant of the coefficients of the left-hand sides of equations \((k')\), and whose numerator is one of the minors of this determinant. By differentiating the expression for \(q_i\), we find the velocity of change of this coordinate

\[ \frac{dq_i}{dt}=\sum_{h=1}^{h=k} n_h C_h^{(i)} E_h \cos(n_h t+\varepsilon_h). \]

From the formulas given, it is seen that each of the coordinates \(q_i\) and each of their derivatives \(q_i'\) is a sum of \(k\) terms, each of which is a periodic function of time.

Depending on the roots of the discriminant \(\Delta(s)\), the periods of the separate constituent parts of each of the sums may either differ from one another, or be partly equal. In either case all \(k\) terms of each sum preserve their independent existence and, by virtue of the difference in the periods and initial phases, or in the phases alone, for different terms of the sum, do not admit of reduction.

Therefore the general formulas for the coordinates and velocities each contain \(2k\) arbitrary constants \(E_h\) and \(\varepsilon_h\) \((h = 1, 2, 3, \ldots, k)\), whose final determination may be made in accordance with the given system of initial positions and velocities of the system.

Without entering into the details of determining the amplitudes and phases from the given initial conditions, which presents no particular features or difficulties, and omitting certain interesting properties of a system performing small oscillations, so as not to go beyond the intended limits, I conclude the present article with a list of the works which I used in writing it.

List

1) Treatise on Natural Philosophy by Lord Kelvin and Tait. Vol. I.
2) The Theory of Sound by Lord Kayleigh. Vol. I.
3) Treatise on the dynamics of a system of rigid bodies by E. D. Routh. Vol. I and II.
4) Weierstrass. Ueber ein die homogenen Functionen zweiten Grades betreffendes Théorem ect. Werke. Bd. I.
5) Jakobi. De binis quibus libet functionibus homogeneis ect. Werke. Bd. III.
H. Weber. Lehrbuch der Algebra. Bd I.
Sylvester. Collected Papers. Vol. I.
Darboux. Mémoire sur la théorie algébrique des formes quadratiques. Liouville Journal. t. 19. 1874.

M. Ivanov.