Consider the differential operators \((m)u=\dfrac{\partial}{\partial x}Mu\), \((\overline m)u=\dfrac{\partial}{\partial y}\left(\dfrac{u}{M}\right)\). Let \(\mathfrak M\) be the set of nonnegative integral powers of the operator \(m\); let \(\overline{\mathfrak M}\) be the set of nonnegative integral powers of the operator \((\overline m)\). The elements of these sets (of either one—indifferently) will be denoted by the letters \(a_1,a_2,\ldots\); the exponent \(a_k\) will be called the exponent of the power \((m)\) or \((\overline m)\) that coincides with \(a_k\).

Introduce \(\otimes\)-multiplication for the \(a_k\), subject to the following rules of calculation. An expression placed in square brackets is subjected to the ordinary action of a differential operator.

1) \((a_k)u=(a_k)[u]\).

2) If \(a_n\in\mathfrak M,\ a_{n-1}\in\mathfrak M\), or \(a_n\in\overline{\mathfrak M},\ a_{n-1}\in\overline{\mathfrak M}\), then
\[ (a_n\otimes a_{n-1}\otimes\cdots\otimes a_1)u =(a_n)[(a_{n-1}\otimes\cdots\otimes a_1)u]. \]

3) If \(a_n\in\mathfrak M,\ a_{n-1}\in\overline{\mathfrak M}\), then
\[ (a_n\otimes a_{n-1}\otimes\cdots\otimes a_1)u =a_n[M^s(a_{n-1}\otimes\cdots\otimes a_1)u], \]
and if \(a_n\in\overline{\mathfrak M},\ a_{n-1}\in\mathfrak M\), then
\[ (a_n\otimes a_{n-1}\otimes\cdots\otimes a_1)u =a_n[M^{-s}(a_{n-1}\otimes\cdots\otimes a_1)u], \]
where \(k_i\) are the exponents of \(a_i\) and \(s=\sum_{i<n}k_i\).

It is known that \(P=\dfrac{\partial^2}{\partial x\,\partial y}\ln M\) is, up to a constant factor, the only integral invariant \((1,1)\). It is also known that any integral invariant of order \(r\) can be represented as a polynomial with real coefficients in invariants of the form \((m^\rho\otimes \overline m^{\sigma})P\) and of orders not exceeding the order of the given invariant.

A necessary and sufficient condition for the nomographability of the equation \(z=\varphi(x,y)\) in a domain \(G\), as is known \({}^{(2)}\), is the compatibility in \(G\) of the Gronwall differential equations
\[ \frac{\partial}{\partial x}\left(2\frac{\partial\overline C}{\partial y}+\frac{\partial\overline D}{\partial x}\right) -\overline C\left(2\frac{\partial\overline C}{\partial y}+\frac{\partial\overline D}{\partial x}\right)=0, \]
\[ \frac{\partial}{\partial y}\left(\frac{\partial\overline C}{\partial y}+2\frac{\partial\overline D}{\partial x}\right) -\overline D\left(\frac{\partial\overline C}{\partial y}+2\frac{\partial\overline D}{\partial x}\right)=0, \]
\[ \overline D=M\overline C+N, \tag{3} \]
where \(\overline C\) is the Gronwall function; \(\overline D\) is its complement; \(M,N\) are expressed by the formulas
\[ M=-\frac{\varphi_y}{\varphi_x},\qquad N=\frac{\partial M}{\partial x}+\frac{1}{M}\frac{\partial M}{\partial y}. \tag{4} \]

The functions
\[ C=\overline C+\frac{\partial}{\partial x}\ln M,\qquad D=\overline D-\frac{\partial}{\partial y}\ln M \tag{5} \]
will be called, respectively, the principal parts of the Gronwall function and of its complement. Obviously, \(D=MC\).

The functions adjoint to the Gronwall function will be called
\[ z_1=\frac13\left(2\frac{\partial\overline C}{\partial y}+\frac{\partial\overline D}{\partial x}\right),\qquad z_2=\frac13\left(\frac{\partial\overline C}{\partial y}+2\frac{\partial\overline D}{\partial x}\right). \tag{6} \]

The Gronwall differential equations can be represented in the form
\[ (m)z_1=MCz_1,\qquad (\overline m)z_2=Cz_2, \]
\[ (m)C=2z_2-z_1-P,\qquad (m)MC=2z_1-z_2+P, \tag{7} \]
where \(P=\dfrac{d^2}{dx\,dy}\ln M\) is the already mentioned Saint-Robert invariant, up to a constant factor.

up to an inessential factor, coinciding with the Blaschke curvature net \((^{1})\).

The coefficients of the polynomial \(\sigma(C,z_1,z_2)\) in \(C,z_1,z_2\) contain the homographic invariants \(T=(1,3)\), \(W=(1,4)\), expressed by the formulas

\[ T=\sum_{\substack{i+k=2\\ i\ge 0,\ k\ge 0}}(m^i\otimes \overline{m}^{\,k})P-3MP^2, \tag{8} \]

\[ W=(m\otimes \overline{m}\otimes m)P+M^3(\overline{m}\otimes m\otimes \overline{m})P +7MP(m)P+M^2P(\overline{m})P, \]

the polynomial \(\sigma(C,z_1,z_2)\) has the form

\[ \sigma(C,z_1,z_2)=M^3PC^3+M\{8MP(z_1-z_2)-T\}C- \{6M^2z_1(\overline{m})P-6Mz_2(m)\}P+W . \tag{9} \]

Theorem 1. The Gronwall system of differential equations, under the condition of sufficient smoothness of \(\varphi(x,y)\), always admits the prolongation

\[ 8MP(m)z_2=\sigma(C,z_1,z_2), \tag{10} \]

\[ (\overline{m})z_1=\frac{1}{M}(m)z_2+(2z_2-2z_1-P)C-\frac{1}{M}(m)P-(\overline{m})P . \tag{11} \]

Proof. Using the identities expressing the commutation rules in double and triple products of the operators \((m)\) and \((\overline{m})\),

\[ M^2(\overline{m}\otimes m)u-(m\otimes \overline{m})u=2MPu, \tag{12^1} \]

\[ M^3(\overline{m}^{\,2}\otimes m)u-(m\otimes \overline{m}^{\,2})u =5M^2P(\overline{m})u+2M^2(\overline{m})P, \tag{12^2} \]

one forms and compares \((m\otimes \overline{m})C\) and \((\overline{m}\otimes m)C\), \((\overline{m}^{\,2}\otimes m)z_1\) and \((m\otimes \overline{m}^{\,2})z_1\). It can be shown that equations (10), (11) exhaust all independent equations relating \(C,z_1,z_2\) and their first derivatives, which follow from (7) and from the commutation rules in triple products of the operators \((m)\) and \((\overline{m})\). For example, comparing \((\overline{m}\otimes m^2)z_2\) and \((m^2\otimes \overline{m})z_2\), one obtains the following consequence of equations (10), (11):

\[ -\frac{8}{M}P(\overline{m})z_1=\sigma_1(C,z_1,z_2), \tag{13} \]

where \(\sigma_1(C,z_1,z_2)\) is symmetric to \(\sigma(C,z_1,z_2)\), i.e. it is obtained from the polynomial \(\sigma(C,z_1,z_2)\) by interchanging \(x,y\).

Theorem 2. If \(C,z_1,z_2\) is a solution of the Gronwall system of differential equations and in the domain \(G\) the Saint-Robert invariant \(P\) is locally different from the identically zero invariant, then \(C,z_1,z_2\) satisfy the system of algebraic equations

\[ \Phi_1(C,z_1,z_2)=3MC^3(m)P-20MP(z_1-2z_2+P)C^2+ 32P(z_1^2+2z_1z_2-2z_2^2)+\cdots=0, \tag{14} \]

\[ \Phi_2(C,z_1,z_2)=3M^2C^3(\overline{m})P+20MP(2z_1-z_2+P)C^2+ 32P(z_2^2+2z_1z_2-2z_1^2)+\cdots=0 \]

(the dots indicate omitted lower-order terms, whose coefficients are homographic invariants \((^{6})\)).

Proof. Equations (7), (10), (11) can be solved with respect to all \((\overline{m})C,\ldots,(\overline{m})z_2\). Forming the necessary integrability conditions, for which one has to use the commutation rule \((12^1)\),

we find what is required. Note that one of the three relations obtained in this way is a trivial identity.

Theorem 3. If, in the domain \(G\), the Saint-Robert invariant \(P\) does not locally vanish identically, then the principal part \(C(x,y)\) of the Gronwall function is found as a common root of a system of polynomials whose coefficients are nomographic invariants.

Theorem 4. The necessary and sufficient conditions for the nomographability of the equation \(z=\varphi(x,y)\) with a sufficiently smooth and monotone right-hand side reduce to two, generally speaking independent, relations connecting the partial derivatives, of order not higher than 9, of the function \(\varphi(x,y)\).

Proof of Theorems 3 and 4. If \(P\) in the domain \(G\) is locally different from identical zero, then equations (14) can be solved with respect to \(z_1,z_2\), which gives an expression for the adjoint functions in terms of \(C,x,y\). Substituting these expressions for \(z_1,z_2\) into the differential equations (7), we find what is required. For \(P\equiv0\), as is known \((^2)\), the equation \(z=\varphi(x,y)\) is always nomographable.

Using invariants, one can in fact write down algebraic equations with respect to \(C\) and the conditions for nomographability; moreover, both the equations for \(C\) and the conditions for nomographability are simplified if it is assumed in advance that one or two scales of the nomogram are rectilinear \((^{8,9})\).

Ivanovo State Pedagogical Institute

Received
5 IV 1957

CITED LITERATURE

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