Theorems on the Most Probable Value of an Adjusted Quantity

V. S. Nuvar’ev

General Considerations

A value of a direct observation, or of a function of direct observations, adjusted by the method of least squares can always be represented in the form of a linear function of the observed quantities, to which there is applicable an important formula, proved by Gauss, Laplace, and later authors, which determines the mean error of this function in the form

\[ m_u^2 = F_x^2 m_x^2 + F_y^2 m_y^2 + \ldots, \tag{1} \]

if

\[ u = F(x, y, \ldots). \]

In formula (1), by \(F_x, F_y, \ldots\) are meant the partial derivatives of \(u\) with respect to \(x, y, \ldots\), and by \(m_x, m_y, \ldots\)—the mean errors of the direct independent observations \(x, y, \ldots\).

The basis of Gauss’s second proof of the method of least squares is the following consideration: if the error equations for determining \(m\) functions of the observations

\[ \varepsilon_i = -l_i + a_i X + b_i Y + c_i Z + \ldots \quad (i = 1,2,\ldots,n) \tag{a} \]

are multiplied by certain positive or negative numbers \(\alpha_i\), subject to the \(m\) conditions

\[ [\alpha a] = 1;\ [\alpha b] = 0;\ [\alpha c] = 0 \ldots, \tag{b} \]

then the true value

\[ X = [\alpha l] + [\alpha \varepsilon] \]

Since the term \([\alpha \varepsilon]\) is unknown, the approximate value \(x\), with error \(u = [\alpha \varepsilon]\), will have the form

\[ x = [\alpha l]. \tag{c} \]

Gauss considers the best determination to be that for which

\[ u^2 = [\alpha \varepsilon]^2 = minimum. \tag{d} \]

By choosing values \(\alpha_i\) satisfying requirement (d) and equations (b), the value \(x\) approaches the true value \(X\).

We constructed the derivations of the theorems on the most probable value of the adjusted quantity sought by us from a consideration opposite to the one described. Namely, we sought first to find the dependence between the true values of the sought and measured quantities. This procedure enabled us to obtain the formulas of the method of least squares directly from the formula for the mean quadratic error of a function of direct observations.

In accordance with the foregoing, we proceeded from the following conditions:

1. The adjusted value of the unknown quantity, irrespective of the form of the function connecting it with the observations, must be equal to its true value if the observations are free of errors.
2. It must be a function of all the observations.
3. The adjusted value must lie in as small a neighborhood as possible of the exact value of the unknown, i.e., the mean error of the adjusted value of the unknown must be minimal.

In our opinion, such a path, from the exact value to the approximate one, has certain logical advantages. As an illustration of what has been said, let us consider the substance of the proof given by Prof. A. S. Chebotarev for the method of conditional observations¹. As is known, the free terms of the conditional equations are determined by the formulas:

\[ w_1 = a_0 + [al];\quad w_2 = b_0 + [bl]\ldots . \]

The basis of the derivation is as follows: “...we shall set ourselves the goal of finding, for the given results of measurements and, consequently, for the given residuals, such values of the coefficients \(k_1, k_2,\ldots,k_r\) for which the function \(u\) will have the smallest mean square error...”

The impression may arise that the multipliers \(k\) depend on the given results of measurements. Undoubtedly, they depend only on the coefficients of the equations and remain constant for any series of measurements.

Let us now consider this proof from another point of view. Suppose that we have found an approximate value \(x^\circ\) satisfying the requirement \(m_x^2=\min\). Clearly, with the same justification we may pose this problem also with respect to the remaining observations. Suppose that, consequently, all adjusted values have been found except the last. But then the adjusted value of this last observation is already determined, since it can be found by substituting \(x^\circ, y^\circ,\ldots\) into any conditional equation (p. 348 of the cited work). This last one may be any of the \(x, y,\ldots\) entering the conditional equations; i.e., a certain difficulty arises: the basis of the proof is the narrow problem of determining the adjusted value of only one observation, but the successive solution of such a problem, owing to the fact that the adjusted values are connected by exact conditions, already gives the last observation a fully determined value. In other words, a proof based on solving the isolated problem of determining the multipliers \(k\), under the condition that the adjusted value be determined with the smallest mean square error, is, in our opinion, not sufficiently rigorous, for example when \(r=1,\ n=2\).

According to the conditions that we have laid as the basis of the derivations, what is sought is the relation between the true values of the unknowns and the quantities subject to measurement. In doing so we proceed as follows: let there be \(r\) exact conditions connecting the true values of the quantities subject to measurement,

\[ \begin{aligned} a_1X_1 + a_2X_2 + a_3X_3 + \cdots + a_0 &= 0,\\ b_1X_1 + b_2X_2 + b_3X_3 + \cdots + b_0 &= 0. \end{aligned} \tag{2} \]

Let us find from equations (2) the true value \(X_1\). Then

\[ X_1^{(1)} = -\frac{1}{a}\left([aX]_1 + a_0\right),\qquad X_1^{(2)} = -\frac{1}{b}\left([bX]_1 + b_0\right)\ldots , \tag{3} \]

where by \([aX]_1\) is meant the sum of the products \(a_iX_i\) without \(a_1X_1\), etc.

According to item 2 of the conditions we have defined, the desired result must be a function of all quantities subject to measurement. This will be achieved if to expressions (3) we add \(X_1 = X_1^{(0)}\). Undoubtedly, \(X_1 = X_1^{(0)} = X_1^{(1)} = X_1^{(2)} = \ldots\). Let us multiply \(X_1\) by \(p_1^{(0)}\), and \(X_1^{(1)}, X_1^{(2)}, \ldots\), respectively, by undetermined factors \(p_1^{(1)}, p_1^{(2)}, \ldots\). Then the true value \(X_1\) can be represented in the form

\[ X_1 = X_1^{(0)} p_1^{(0)} + [X_1^{(j)} p_1^{(j)}], \quad (j = 1, 2, \ldots, r) \tag{4} \]

The last equation has meaning only under the condition

\[ [p_1^{(s)}] = p_1^{(0)} + [p_1^{(j)}] = 1 \tag{5} \]

One could represent \(X_1\) in the form

\[ X_1 = \frac{X_1^{(0)} p_1^{(0)} + [X_1^{(j)} p_1^{(j)}]}{r+1}, \tag{6} \]

but then

\[ [p_1^{(s)}] = r + 1 \tag{7} \]

Thus, from equation (4) it follows that

\[ X_1 = F(X_i^{(s)}p_i^{(s)}); \qquad (i = 1,2\ldots n;\ s = 0,1,2\ldots r) \tag{8} \]

Expression (8) gives the true value \(X_1\) for any values \(p_1^{(s)}\) satisfying equation (5). If in (3), instead of the true values \(X_i\), the measured values \(l_i\) are substituted, then equation (8) will give an approximate value \(x_1\), depending on the choice of the factors \(p_1^{(s)}\).

Let us find the set of values \(p_1^{(s)}\) under the condition

\[ m_{x_1}^{2} = minimum. \tag{9} \]

Suppose that \(X_1, X_2, \ldots\) will be found from measurements with mean errors \(m_{x_1}, m_{x_2}, \ldots\). Then, in order to solve the problem posed, it is necessary to write the expression

\[ m_{x_1}^{2} = f(m_{x_1}, m_{x_2}, \ldots, m_{x_r}, p_1^{(0)}, p_1^{(1)}, \ldots, p_1^{(r)}) \tag{10} \]

and equate the partial derivatives of equation (10) with respect to \(p_1^{(s)}\) to the partial derivatives with respect to \(p_1^{(s)}\) of equation (5), multiplied by \(2k_1\). The resulting \(r+1\) equations, together with equation (5), will make it possible to find the \(r+1\) factors \(p_1^{(s)}\) and the correlate \(k_1\). Substituting \(p_1^{(s)}\) into equation (4), we, in view of equalities (3), obtain the true value \(X_1\) as a function of the true values of all quantities \(l_i\) subject to measurement and of the factors \(p_i^{(s)}\).

Let us note that we have proceeded only from the system of conditional equations (2). Expression (4) can be written for each value \(X_i\), and substitution of \(X_i\) \((i = 1, 2, \ldots, n-1)\) into equations (2) does not affect the value of \(X_n\), thereby eliminating the difficulty indicated above. In this we see the advantages of the method of reasoning “from the exact to the approximate.”

Theorem on the most probable value of direct observations connected by \(r\) exact conditions

Theorem 1. The most probable value of each of \(n\) observed quantities connected by \(r\) exact conditions \((n > r)\) can be represented in the form of the arithmetic mean of the result of the direct measurement of the desired quantity and its dependent values obtained from \(r\) simultaneous solutions of equations, or delivered by each conditional equation, with weights subordinated to the requirement of the greatest weight of the desired quantity.

First proof. Use of values obtained from each conditional equation¹

For simplicity of exposition, take two conditional equations with four unknowns²

\[ \begin{aligned} a_1X_1+a_2X_2+a_3X_3+a_4X_4+a_0&=0\\ b_1X_1+b_2X_2+b_3X_3+b_4X_4+b_0&=0. \end{aligned} \tag{11} \]

Solve each of equations (11) with respect to \(X_1\); then, if the observations were free of errors, the true value of \(X_1\) could be represented in the following form:

\[ X_1^{(0)}=X_1 \]

\[ X_1^{(1)}=-\frac{1}{a_1}\bigl([aX]_1+a_0\bigr) \tag{12} \]

\[ X_1^{(2)}=-\frac{1}{b_1}\bigl([bX]_1+b_0\bigr). \]

Multiplying the equalities (12) by multipliers \(p_1^{(0)}, p_1^{(1)}, p_1^{(2)}\), satisfying the condition

\[ [p_1^{(s)}]=1,\qquad (s=0,1,2), \tag{13} \]

the true value \(X_1=X_1^{(0)}\) can be represented in the form

\[ X_1=X_1^{(0)}p_1^{(0)}+X_1^{(1)}p_1^{(1)}+X_1^{(2)}p_1^{(2)}. \tag{14} \]

If in the right-hand side of equations (12) we substitute the measured values \(l_i\), then, on the basis of (14), we obtain an approximate value \(x_1\) of the quantity \(X_1\)

\[ x_1=l_1p_1^{(0)}-\left(\frac{a_2}{a_1}p_1^{(1)}+\frac{b_2}{b_1}p_1^{(2)}\right)l_2 -\left(\frac{a_3}{a_1}p_1^{(1)}+\frac{b_3}{b_1}p_1^{(2)}\right)l_3- \]

\[ -\left(\frac{a_4}{a_1}p_1^{(1)}+\frac{b_4}{b_1}p_1^{(2)}\right). \tag{15} \]

Suppose that all observations have equal weight³; then, putting \(m=1\), we shall have

\[ m_{x_1}^{2}=(p_1^{(0)})^{2}+\frac{[a^{2}]}{a_1^{2}}(p_1^{(1)})^{2} +\frac{[b^{2}]}{b_1^{2}}(p_1^{(2)})^{2} +2\frac{[a_i b_i]}{a_1b_1}p_1^{(1)}p_1^{(2)} \tag{16} \]

Let \(m_{x_1}^{2}=\textit{minimum}\).

Then, multiplying equation (13) by \(2k_1\), differentiating \(2k_1[p_1^{(s)}]\) and equation (16) with respect to the variables \(p_1^{(s)}\), and equating the differentials with respect to each variable, we obtain:

\[ \frac{[a^{2}]_1}{a_1^{2}}p_1^{(1)}+\frac{[ab]_1}{a_1b_1}p_1^{(2)}=k_1 \]

\[ \frac{[ab]_1}{a_1b_1}p_1^{(1)}+\frac{[b^{2}]_1}{b_1^{2}}p_1^{(2)}=k_1, \tag{17} \]

whence

\[ p_1^{(0)}=k_1;\qquad p_1^{(1)}=k_1\frac{A_1}{D};\qquad p_1^{(2)}=k_1\frac{A_2}{D} \]

\[ A_1=\frac{[b^2]_1}{b_1^2}-\frac{[ab]_1}{a_1b_1}; \qquad A_2=\frac{[a^2]_1}{a_1^2}-\frac{[ab]_1}{a_1b_1}; \qquad D=\frac{[a^2]_1[b^2]_1-[ab]_1^2}{a_1^2b_1^2}. \tag{18} \]

Substituting the values of \(p_1^{(s)}\) into equation (13), on the basis of (18), we obtain

\[ k_1=\frac{D}{D+A_1+A_2} \]

\[ p_1^{(0)}=\frac{D}{D+A_1+A_2};\qquad p_1^{(1)}=\frac{A_1}{D+A_1+A_2};\qquad p_1^{(2)}=\frac{A_2}{D+A_1+A_2}. \tag{19} \]

Thus we have found expressions \(p\), having the meaning of weights, with which the result of the direct measurement and its dependent values obtained from each conditional equation enter into the arithmetic mean (14).

Let us show that the expression of the value \(x_1\) adjusted according to formulas (12)—(19) can be reduced to the usual derivation of \(x_1^0\), obtained through correlates.

Add to the right-hand side of the second and third equations (12) \(0=X_1-X_1\); then for the true values \(X_1^{(s)}\) we shall have the following expressions:

\[ X_1^{(0)}=X_1, \]

\[ X_1^{(1)}=X_1-\frac{1}{a_1}([aX]+a_0) \]

\[ X_1^{(2)}=X_1-\frac{1}{b_1}([bX]+b_0). \]

Multiplying these values by the factors \(p_1^{(s)}\), subject to conditions (13) and \(m_x^2=\min\), and replacing in the right-hand side \(X_i\) by \(l_i\), we obtain

\[ x_1^0=l_1-\frac{w_1}{a_1}p_1^{(1)}-\frac{w_2}{b_1}p_1^{(2)}, \]

where \(w_1, w_2\) are the constant terms of the conditional equations (11). Then the correction to the observed value \(l_1\) will be equal to

\[ v_1=x_1^0-l_1=-\left(\frac{w_1}{a_1}p_1^{(1)}+\frac{w_2}{b_1}p_1^{(2)}\right) \tag{20} \]

On the basis of (18) and (19) we have

\[ D+A_1+A_2= \frac{[a^2]_1[b^2]_1-[ab]_1^2+[b^2]_1a_1^2+[a^2]_1b_1^2-2[ab]_1a_1b_1}{a_1^2b_1^2} \]

Add to the numerator the identity \(0=a_1^2b_1^2-a_1^2b_1^2\); then

\[ D+A_1+A_2 =\frac{1}{a_1^2b_1^2}([aa][bb]-[ab]^2) =\frac{1}{a_1^2b_1^2}D_0. \tag{21} \]

It is known that \(D_0\) is the determinant of the system of normal equations corresponding to the conditional equations (11).

Taking into account (18), (19), and (21), we find the value of the correction \(v_1\) from (20) in the form

\[ v_1=-\frac{1}{D_0}\left\{\frac{w_1}{a_1}\left([b^2]a_1^2-[ab]a_1b_1\right)+\frac{w_2}{b_1}\left([a^2]b_1^2-[ab]a_1b_1\right)\right\}. \]

Let us add to each expression in parentheses \(0=a_i^2b_1^2-a_1^2b_i^2\); then

\[ v_1=-\frac{1}{D_0}\left\{w_1\left([b^2]a_1-[ab]b_1\right)+w_2\left([a^2]b_1-[ab]a_1\right)\right\}. \tag{22} \]

But formula (22) is the expression for the correction found through the correlates, which proves the possibility of using the arithmetic mean of the direct measurement of the unknown quantity and of its indirect values, found from each conditional equation, with weights satisfying the requirement of the greatest weight of the unknown.

Let us apply the formulas found to the consideration of a particular example.

Suppose there is a system of conditional equations

\[ \begin{aligned} -4X_1+X_2+2X_3+X_4+a_0&=0\\ 5X_1-2X_2+X_3+2X_4+b_0&=0 \end{aligned} \tag{23} \]

Assume that the values \(X_i\) are equal to:

	\(X_1\)	\(X_2\)	\(X_3\)	\(X_4\)
True \(X_i\)	6	8	4	9
Measured \(l_i\)	5	7	4	11

Then \(a_0=-1\); \(b_0=-36\); \(w_1=[al]+a_0=5\); \(w_2=1\).

Solving system (23) by the method of least squares, we find:

\(v_1=+0.45\), i.e. the adjusted value

\[ x_1^0=5.45 \]

with mean error \(m_{x_0}^2=0.118\).

Let us apply to this same example the formulas derived by us.

Substituting into equations (12) the data of the direct measurements, we obtain the following values for \(x_1\):

\[ l_1=5;\qquad x_1^{(1)}=-\frac{1}{a_1}\left([al]_1+a_0\right)=6.25;\qquad x_1^{(2)}=-\frac{1}{b_1}\left([bl]_1+b_0\right)=4.80. \]

By formulas (18) and (19), the values \(p_1^{(s)}\) are equal to:

\[ p_1^{(0)}=0.118;\qquad p_1^{(1)}=0.434;\qquad p_1^{(2)}=0.448 \]

The adjusted value according to formula (14) is \(x_1^0=5.425\).

Substituting \(p_1^{(s)}\) into (16), we find

\[ m_x^2=(0.118)^2+\frac{6}{16}(0.434)^2-\frac{9}{25}(0.448)^2-\frac{2}{10}(0.434\times0.448)=0.1181, \]

i.e., we obtain the previous result.

In the present case we sought the adjusted value \(x_1^0\). However, one may use these same formulas by posing the problem of determining the correction to the direct measurement. The adjusted value \(x_1^0\) is found from (14):

\[ x_1^0 = l_1 p_1^{(0)} + x_1^{(1)} p_1^{(1)} + x_1^{(2)} p_1^{(2)} . \tag{24} \]

The expressions (12) for \(X_1^{(1)}\) and \(X_1^{(2)}\) can be transformed by adding to the right-hand sides \(0=X_1-X_1\). Then the equalities (12) take the following form:

\[ X_1^{(0)} = X_1, \]

\[ X_1^{(1)} = X_1 - \frac{1}{a_1}\bigl([aX]+a_0\bigr), \]

\[ X_1^{(2)} = X_1 - \frac{1}{b_1}\bigl([bX]+b_0\bigr). \]

If, in the right-hand sides, we replace the true value \(X_1\) by the observed value \(l_1\), we obtain:

\[ x_1=l_1;\quad x_1^{(1)}=l_1-\frac{w_1}{a_1};\quad x_1^{(2)}=l_1-\frac{w_2}{b_1}. \tag{25} \]

On the basis of (24) and (13),

\[ v_1=x_1^0-l_1=-\left(\frac{w_1}{a_1}p_1^{(1)}+\frac{w_2}{b_1}p_1^{(2)}\right). \tag{26} \]

Since in the present example \(w_1=5,\ w_2=1\), we have

\[ v_1=-\left(-\frac{5}{4}\,0.434+\frac{1}{5}\,0.448\right)=0.452, \]

which agrees with the correction found by means of correlates. It is known that the adjusted value \(x_i^{(0)}\) can be expressed in terms of the transition coefficients \(r_j\) in the following form:

\[ x_i^0=l_i+[rw]. \]

Comparing this expression with (26), we find that the weights with which the values \(x_i^{(s)}\), obtained from each conditional equation, enter into the arithmetic mean are equal to

\[ p_i^{(1)}=-a_i r_1;\qquad p_i^{(2)}=-b_i r_2. \tag{27} \]

Second proof. Use of the values obtained from the joint solution of the conditional equations

Let us apply the considerations presented in the preceding paragraph to the construction of a proof of the first theorem in the case when the dependent values of the sought quantity are obtained from the joint solution of the conditional equations.

Let us find the value \(X_1\) from the joint solution\(^1\) of equations (11); for this purpose we first solve the system (11) with respect to \(x_1\) and \(x_2\); then

\[ X_{12}=\left\{(a_2b_3-b_2a_3)X_3+(a_2b_4-a_4b_2)X_4+(a_2b_0-b_2a_0)\right\}\frac{1}{D_{12}}. \]

\(^1\) Such a solution is possible if the rank of the matrix

\[ A= \begin{pmatrix} a_1 & a_2 & a_3 & a_4\\ b_1 & b_2 & b_3 & b_4 \end{pmatrix} \]

is equal to the rank of the matrix

\[ B= \begin{pmatrix} a_1 & a_2 & a_3 & a_4 & a_0\\ b_1 & b_2 & b_3 & b_4 & b_0 \end{pmatrix}. \]

Exactly similarly, solving the systems (11) with respect to \(X_1\) and \(X_3\), we obtain:

\[ X_{13}=\frac{1}{D_{13}}\left\{(a_3b_2-b_3a_2)X_2+(a_3b_4-b_3a_4)X_4+(a_3b_0-b_3a_0)\right\}. \]

Then the left-hand sides of equality (28) represent the true values of \(X_1\), if in the right-hand side under \(X_i\) one understands the true values

\[ \begin{aligned} X_1 \\ X_{12}&=\frac{1}{D_{12}}\left(D_{23}X_3+D_{24}X_4+D_{20}\right) \\ X_{13}&=\frac{1}{D_{13}}\left(-D_{23}X_2+D_{34}X_4+D_{30}\right) \end{aligned} \tag{28} \]

Multiplying the expressions (28), respectively, by the factors \(p_1,p_2,p_3\), then the true value \(X_1\) will take the form

\[ X_1=X_1p_1+X_{12}p_2+X_{13}p_3, \tag{29} \]

if the factors \(p_1, p_2, p_3\) are made to satisfy the condition

\[ p_1+p_2+p_3=1. \tag{30} \]

On the basis of (29) and (28) we have

\[ X_1=p_1X_1-\frac{D_{23}}{D_{13}}p_3X_2+\frac{D_{23}}{D_{12}}p_2X_3+ \left(\frac{D_{24}}{D_{12}}p_2+\frac{D_{34}}{D_{13}}p_3\right)X_4+p_2\frac{D_{20}}{D_{12}}+ \]

\[ +p_3\frac{D_{30}}{D_{13}}. \tag{31} \]

Thus we have represented \(X_1\) in the form of a linear function of quantities subject to measurement. If the measured values of the quantities \(X_i\) are substituted into the right-hand side of equation (31), then equation (31) will determine an approximate value \(x_1\) with a mean square error depending on the system of factors \(p_i\). The most advantageous value of \(x_1\) will be the value for which it deviates minimally from \(X_1\), i.e. when

\[ m_{x_1}^2=minimum; \]

for the case of equally accurate values; hence, for \(m^2=1\),

\[ m_{x_1}^2=p_1^2+\frac{D_{23}^2+D_{24}^2}{D_{12}^2}p_2^2+ \frac{D_{23}^2+D_{34}^2}{D_{13}^2}p_3^2+ 2\frac{D_{24}D_{34}}{D_{12}D_{13}}p_2p_3=min \tag{32} \]

Multiply the conditional equation (30) by the as yet undetermined factor \(2k\), and equate the partial derivatives of equation (32) with respect to the variables \(p_i\) to the partial derivatives of the expression \(2k[p]\) with respect to these same variables. Then

\[ p_1=k; \]

\[ \frac{1}{D_{12}^2}\left(D_{23}^2+D_{24}^2\right)p_2+ \frac{D_{24}D_{34}}{D_{12}D_{13}}p_3=k \]

\[ \frac{D_{24}D_{34}}{D_{12}D_{13}}p_2+ \frac{1}{D_{13}^2}\left(D_{23}^2+D_{34}^2\right)p_3=k. \tag{33} \]

Solving the last two equations gives

\[ p_2=\frac{1}{D_0}\left\{\frac{1}{D_{13}^{2}}\left(D_{23}^{2}+D_{34}^{2}\right)-\frac{D_{24}D_{34}}{D_{12}D_{13}}\right\}k=\frac{D_2}{D_0}k \]

\[ p_3=\frac{1}{D_0}\left\{\frac{1}{D_{12}^{2}}\left(D_{23}^{2}+D_{24}^{2}\right)-\frac{D_{24}D_{34}}{D_{12}D_{13}}\right\}k=\frac{D_3}{D_0}k \]

Substituting the values of \(p_i\) in (30), we have:

\[ k=\frac{D_0}{D_0+D_2+D_3}, \]

therefore

\[ p_1=\frac{D_0}{N} \]

\[ p_2=\frac{1}{N}\left\{\frac{1}{D_{13}^{2}}\left(D_{23}^{2}+D_{34}^{2}\right)-\frac{D_{24}D_{34}}{D_{12}D_{13}}\right\} \]

\[ p_3=\frac{1}{N}\left\{\frac{1}{D_{12}^{2}}\left(D_{23}^{2}+D_{24}^{2}\right)-\frac{D_{24}D_{34}}{D_{12}D_{13}}\right\}, \]

where \(D_0\) is the determinant of system (33), and \(N=D_0+D_2+D_3\).

Let us find the values of \(D_0\) and \(N\).

\[ D_0=\frac{1}{D_{12}^{2}D_{13}^{2}}\left(D_{23}^{2}+D_{24}^{2}\right)\left(D_{23}^{2}+D_{34}^{2}\right)-\frac{D_{24}^{2}D_{34}^{2}}{D_{12}^{2}D_{13}^{2}} \]

\[ D_0=\frac{D_{23}^{2}}{D_{12}^{2}D_{13}^{2}}\left(D_{23}^{2}+D_{24}^{2}+D_{34}^{2}\right) \]

\[ N=\frac{1}{D_{12}^{2}D_{13}^{2}}\left\{\left(D_{12}^{2}+D_{13}^{2}+D_{23}^{2}+D_{24}^{2}+D_{34}^{2}\right)D_{23}^{2}+\left(D_{12}D_{34}-D_{13}D_{24}\right)^{2}\right\}. \]

but

\[ D_{12}D_{34}-D_{13}D_{24}=-D_{23}D_{14} \tag{34} \]

or

\[ N=\frac{D_{23}^{2}D}{D_{12}^{2}D_{13}^{2}}, \]

where

\[ D=D_{12}^{2}+D_{13}^{2}+D_{14}^{2}+D_{23}^{2}+D_{24}^{2}+D_{34}^{2}. \tag{35} \]

As we shall see later, \(D\) is the determinant of the system of normal equations corresponding to the given system of conditional equations.

Using the values of \(D_0\) and \(N\), we transform the expressions \(p_i\):

\[ p_1=\frac{D_{23}^{2}+D_{24}^{2}+D_{34}^{2}}{D}; \]

\[ p_2=\frac{D_{12}^{2}D_{13}^{2}}{D\,D_{23}^{2}}\left\{\frac{1}{D_{13}^{2}}\left(D_{23}^{2}+D_{34}^{2}\right)-\frac{D_{24}D_{34}}{D_{12}D_{13}}\right\}; \]

\[ p_3=\frac{D_{12}^{2}D_{13}^{2}}{D\,D_{23}^{2}}\left\{\frac{1}{D_{12}^{2}}\left(D_{23}^{2}+D_{24}^{2}\right)-\frac{D_{24}D_{34}}{D_{12}D_{13}}\right\}. \]

Carrying out the multiplication and taking (35) into account, we find the values of \(p_i\) in final form:

\[ p_1=\frac{1}{D}\left(D_{23}^{2}+D_{24}^{2}+D_{34}^{2}\right); \]

\[ p_2=-\frac{D_{12}}{D D_{23}}\left(D_{12}D_{23}-D_{14}D_{34}\right); \tag{36} \]

\[ p_3=-\frac{D_{13}}{D D_{23}}\left(D_{13}D_{23}+D_{14}D_{24}\right). \]

Formulas (36) solve the problem of finding the multipliers, having the meaning of weights, with which the directly measured value \(l_1\) and the values \(x_{12}\) and \(x_{13}\), obtained from the solution of the conditional equations, enter into the arithmetic mean (29). Here \(p_i\) satisfies the conditions: \(m_{x_1}=\min\) and \([p]=1\). If we had taken the case of measurements of unequal accuracy, then the mean square error (32) would have the form

\[ m_{x_1}^{2}=p_1^{2}m_{x_1}^{2}+\frac{D_{23}^{2}m_{x_3}^{2}+D_{24}^{2}m_{x_4}^{2}}{D_1^{2}}p_2^{2}+\ldots \]

i.e., the partial derivatives of \(m_{x_1}^{2}\) with respect to \(p_i\) would change, but the general course of the solution would remain the same.

We have considered the question of finding the multipliers \(p_i\), which determine the adjusted value \(x\) with the least mean square error, or with the greatest weight. In this derivation we proceeded from the mean square error of a composite deviation that is a linear function of the individual deviations, which obey the normal law of distribution.

Let us show that the results of adjustment by formulas (31)—(36) and by the method of conditional observations of the method of least squares coincide, i.e., that the adjusted values of the measured quantities in both cases will be equal.

From the general theory of the method of least squares it is known that the adjusted value \(x_1^0\) of the measured quantity \(x_1\) can be represented in the form:

\[ x_1^0=(1+a_1r_1+b_1r_2+\ldots)l_1+(0+a_2r_1+b_2r_2+\ldots)l_2+\ldots+a_0r_1+ \]

\[ +b_0r_2\ldots \]

or, introducing the notations

\[ F_1=1+a_1r_1+b_1r_2+\ldots \qquad F_2=0+a_2r_1+b_2r_2+\ldots, \]

we obtain:

\[ x_1^0=\sum F_i l_i+a_0r_1+b_0r_2, \tag{37} \]

where \(r_1,\ r_2\ldots\) are transitional coefficients determined by the equations:

\[ \left. \begin{aligned} [aa]r_1+[ab]r_2+\ldots+a_1&=0\\ [ab]r_1+[bb]r_2+\ldots+b_1&=0. \end{aligned} \right\} \tag{38} \]

From formula (31) it follows that

\[ x_1^0=p_1l_1-\frac{D_{23}}{D_{13}}p_3l_2+\ldots\ldots+p_2\frac{D_{20}}{D_{12}}+p_3\frac{D_{30}}{D_{13}}, \]

i.e., we shall solve the posed problem if we prove that

\[ F_1=p_1;\qquad F_2=-\frac{D_{23}}{D_{13}}p_3;\qquad F_3=\frac{D_{23}}{D_{12}}p_2;\qquad F_4=\frac{D_{24}}{D_{12}}p_2+\frac{D_{34}}{D_{13}}p_3; \tag{39} \]

\[ a_0r_1+b_0r_2=\frac{D_{20}}{D_{12}}p_2+\frac{D_{30}}{D_{13}}p_3 . \]

In this case the values of \(x_1^0\) according to formula (37) and according to formula (31) coincide.

Let us transform equation (37), which determines the value \(x_1^0\) adjusted by the method of least squares, into another form.

The transition coefficients \(r_1\) and \(r_2\) from (38) are equal to

\[ r_1=\frac{1}{D}\left([ab]b_1-[bb]a_1\right) \]

\[ r_2=\frac{1}{D}\left([ab]a_1-[aa]b_1\right). \tag{40} \]

Then

\[ F_1=\frac{1}{D}\left(D+a_1b_1[ab]-a_1^2[bb]+a_1b_1[ab]-b_1^2[aa]\right). \]

Transform the expression standing in parentheses. After multiplication, collecting like terms and omitting the details, we obtain the expression in parentheses in the form

\[ D_{23}^2+D_{24}^2+D_{34}^2 . \tag{41} \]

On the same grounds

\[ D=D_{12}^2+D_{13}^2+D_{14}^2+D_{23}^2+D_{24}^2+D_{34}^2 \]

or

\[ F_1=\frac{D_{23}^2+D_{24}^2+D_{34}^2}{D} \tag{42} \]

From (37) and (40) we find

\[ F_2=\frac{1}{D}\left\{a_2b_1[ab]-a_1a_2[bb]-a_1b_2[ab]-b_1b_2[aa]\right\}. \]

After the transformations of the expression \(F_1\) indicated above, for \(F_2\), \(F_3\), and \(F_4\) we obtain the following values:

\[ F_2=-\frac{1}{D}\left(D_{13}D_{23}+D_{14}D_{24}\right); \]

\[ F_3=\frac{1}{D}\left(D_{12}D_{23}-D_{14}D_{34}\right); \tag{43} \]

\[ F_4=\frac{1}{D}\left(D_{12}D_{24}+D_{13}D_{34}\right). \]

Now let us pass to the last terms of the right-hand side of equation (37). After substituting the values of \(r_1, r_2\) and carrying out the necessary operations, \(a_0r_1+b_0r_2\) is reduced to the form

\[ a_0r_1+b_0r_2=\frac{1}{D}\left(D_{20}D_{12}+D_{30}D_{13}+D_{40}D_{14}\right). \tag{44} \]

Here it is necessary to recall that \(D_{ik}\) is the determinant of the system of two conditional equations when they are solved with respect to the unknowns numbered \(i\) and \(k\). The symbol \(D_{i0}\) (by analogy with \(D_{ik}\)) may be called the determinant of the same system, formed from the coefficients of \(x_i\) and the absolute terms \(a_0\) and \(b_0\) entering the system of conditional equations (11).

Let us now turn to formula (31), which determines the adjusted value \(x_1^0\) under the condition

\[ m_{x_1}^{2} = minimum. \]

If in (31) \(X_i\) were replaced by the measured values \(l_i\), then the conditions for the coincidence of the results of adjustment by formula (31) and by the formulas of the method of conditional observations would be the equalities (39).

From the comparison of (42) and (36) we immediately find that

\[ F_1 = p_1. \]

We shall also prove that the coefficient of \(X_2\) in equation (31) will be equal to \(F_2\) in equation (37), i.e.

\[ -\frac{D_{23}}{D_{13}}p_3 = F_2 = ? \]

On the basis of (31) and (36) we have:

\[ -\frac{D_{23}}{D_{13}}p_3 = -\frac{1}{D}(D_{13}D_{23}+D_{14}D_{24}). \tag{45} \]

In exactly the same way,

\[ -\frac{D_{23}}{D_{12}}p_2 = \frac{1}{D}(D_{12}D_{23}-D_{14}D_{34}); \tag{46} \]

\[ \frac{D_{24}}{D_{12}}p_2+\frac{D_{34}}{D_{13}}p_3 = \frac{D_{24}}{D_{23}D}(D_{12}D_{23}-D_{14}D_{34}) + \frac{D_{34}}{D_{23}D}(D_{13}D_{23} + D_{14}D_{24}) = \frac{1}{D}(D_{12}D_{24}+D_{13}D_{34}). \tag{47} \]

Comparing (45), (46), (47), and (48), we find that the coefficients of \(l_i\) in equations (31) and (37) are identical.

Let us transform the last two terms of equation (31).

\[ Q = \frac{D_{20}}{D_{12}}p_2 + \frac{D_{30}}{D_{13}}p_3 = \frac{D_{20}}{D_{23}D}(D_{12}D_{23}-D_{14}D_{31}) + \frac{D_{30}}{D_{23}D}(D_{13}D_{23}+D_{14}D_{24}); \]

\[ Q = \frac{1}{D} \left\{ D_{20}D_{12} + D_{30}D_{13} + \frac{D_{14}}{D_{23}}(D_{30}D_{24}-D_{20}D_{34}) \right\}; \]

\[ D_{30}D_{24}-D_{20}D_{34} = (a_3b_0-b_3a_0)(a_2b_4-a_4b_2) - (a_2b_0-b_2a_0)(a_3b_4-a_4b_3) = \]

\[ = -a_2b_3b_4a_0-a_3a_4b_2b_0+a_3b_2b_4a_0+a_2a_4b_3b_0 = \]

\[ = (a_2b_3-a_3b_2)(a_4b_0-b_4a_0) = D_{23}D_{40}. \]

Substituting into \(Q\) the obtained value of the bracket, we have:

\[ Q = \frac{D_{20}}{D_{12}}p_2 + \frac{D_{30}}{D_{13}}p_3 = \frac{1}{D}(D_{20}D_{12}+D_{30}D_{13}+D_{40}D_{14}). \tag{48} \]

We note that expressions (48) and (44) coincide, i.e., equalities (39), confirming the identity of the results of adjustment under the condition of the minimum of the sum of squares of the corrections and by the formulas of Theorem 1, have been proved. In this proof we have omitted cumbersome algebraic transformations.

Consider expression (31)

\[ X_1=p_1X_1-\frac{D_{23}}{D_{13}}p_2X_2+\ldots+p_2\frac{D_{20}}{D_{12}}+p_3\frac{D_{30}}{D_{13}}, \]

where \(D_{12}, D_{\ldots}, \ldots, D_{30}\) are determinants of the system of conditional equations

\[ [a_iX_i]+a_0=0 \]

\[ [b_iX_i]+b_0+0, \]

if from them one determines, respectively, \(X_1, X_2,\ldots X_2, X_3,\ldots X_3, a_0, b_0\) (the symbols \(D_{20}, D_{30}\) have been introduced for symmetry), while the \(p_i\) are subject to the equation \([p]=1\).

We have proved that \(x_1^0\) from (31) will have a value coinciding with the value \(x_1^0\) adjusted by the method of least squares, if

\[ m_{x_1^0}^{\,2}=\min. \]

Substituting in (28) and (29), instead of the true values \(X_i\), the measured values \(l_i\), we obtain the values \(x_1, x_{12}, x_{13}\)

\[ x_1=l_1;\quad x_{12}=\frac{1}{D_{12}}\left(D_{23}l_3+D_{21}l_1+D_{20}\right);\quad x_{13}=\frac{1}{D_{13}}\left(-D_{21}l_2+D_{34}l_1+D_{30}\right), \tag{a} \]

and the value \(x_1^0\), adjusted by the method of least squares,

\[ x_1^0=x_1p_1+x_{12}p_2+x_{13}p_3, \tag{b} \]

where the system \(p_i\) satisfies the conditions \([p]=1\) and

\[ m_{x_1^0}^{\,2}=minimum \]

From consideration of expressions (a) and (b) it follows that \(x_1^0\) is a general arithmetic mean of dependent quantities, since the same measured quantity \(l_1\) enters into the determination of the values \(x_{12}\) and \(x_{13}\).

The same conclusion can be drawn from formulas (12) and (14).

Example of adjustment by formulas (29)—(36)

Take the system of conditional equations

\[ -4X_1+X_2+2X_3+X_4+a_0=0, \]

\[ 5X_1-2X_2+X_3+2X_4+b_0=0. \]

Suppose that the true values \(X_i\) are respectively equal to \(6, 8, 4, 9\); then

\[ -4X_1+X_2+2X_3+X_4-1=0, \]

\[ 5X_1-2X_2+X_3+2X_4-36=0. \tag{a} \]

Equations (38) for determining the transition coefficients will have the form

\[ \begin{array}{r} +22r_1-18r_2-4=0\\ -18r_1+34r_2+5=0 \end{array} \left|\quad r_1=\frac{23}{212};\qquad r_2=-\frac{19}{212}. \right. \]

The value \(x_1^0\), adjusted by the method of least squares through the transition coefficients (formula 37), can be expressed as follows:

\[ x_1^0 = [Fl] + a_0 r_1 + b_0 r_2, \]

where

\[ F_1 = 1 + a_1 r_1 + b_1 r_2; \qquad F_2 = 0 + a_2 r_1 + b_2 r_2,\ldots \]

For clarity, let us assign observed values \(l_i\) and compute the adjusted value \(x_1^0\) through \(F_i\). Let

	\(a\)	\(b\)	\(L_i\)	\(l_i\)	\(al\)	\(bl\)	\(F_i\)	\(F_i l_i\)
1	\(-4\)	5	6	5	\(-20\)	25	0,120	0,600
2	1	\(-2\)	8	7	7	\(-14\)	0,288	2,015
3	2	1	4	4	8	4	0,127	0,508
4	1	2	9	11	11	22	\(-0,071\)	\(-0,781\)
					6	37		2,347

\[ a_0 r_1 + b_0 r_2 = 3,118 \]

\[ x_1^0 = 5,46 \]

\[ m_{x_1}^{2} = [F^2] = 0,1184 \]

Let us solve this same problem through correlates.

The free terms of the conditional equations will be equal to

\[ w_1 = [al] + a_0 = 5, \qquad w_2 = [bl] + b_0 = 1. \]

Then

\[ \begin{aligned} 22 k_1 - 18 k_2 + 5 &= 0,\\ 18 k_1 + 34 k_2 + 1 &= 0 \end{aligned} \qquad \left| \quad k_1 = -\frac{188}{424}; \qquad k_2 = -\frac{112}{424} \right. \]

\[ v_1 = +0,45; \qquad x_1^0 = 5,45. \]

Now let us apply the formulas derived by us in the second proof of theorem 1. From formula (28) it follows that the solution of system \((a)\) gives the following values of \(X_i\):

\[ X_{12} = \frac{1}{D_{12}}\left(D_{23}X_3 + D_{24}X_4 + D_{20}\right) \]

\[ X_{13} = \frac{1}{D_{13}}\left(-D_{23}X_2 + D_{34}X_4 + D_{30}\right) \tag{b} \]

Let us compute the determinants of system \((a)\)

	\(D_{12}\)	\(D_{13}\)	\(D_{14}\)	\(D_{23}\)	\(D_{24}\)	\(D_{34}\)	\([D^2]\)
\(D_i\)	3	\(-14\)	\(-13\)	5	4	3	—
\(D_i^2\)	9	196	169	25	16	9	424

Let us first set ourselves the goal of determining the true value \(X\) satisfying the first and second conditions (general considerations). To this end, from the system (49) we shall choose, say, the first \(m\) equations and find the value \(X_1\) of the unknown \(X\) from their algebraic solution. Then

\[ X_1=f_1(L_i);\qquad (i=1,2,\ldots,m). \tag{50} \]

But the same value can be obtained by discarding one of the first \(m\) equations and adding one, say the \(m+1\)-st equation. Then the true value \(X\) can be represented in the form

\[ X_2=f_2(L_i);\qquad (i=1,2,\ldots,m-1,m+1) \tag{51} \]

Let us stipulate in advance that the result of the equation does not depend on which equations enter each combination. The only condition for each new combination is the restriction that it include one equation not previously used. Obviously, under this condition the number of all combinations will be \(n-m+1\).

Below we shall show that \(n-m+1\) combinations are necessary and sufficient. The use of all combinations (by Jacobi’s theorem) or even \(n-m+2\) combinations makes the determinant of the system of equations (63) vanish, i.e., these equations have more than one system of solutions.

Let us show that from equations (49) one can always obtain \(n-m+1\) necessary solutions. This follows from the proof of the proposition: in a matrix with \(n\) rows and \(m\) columns (\(n>m\)), each of whose \(m-1\) rows has no linear dependence on the other rows of the matrix, the number of nonzero determinants of order \(m\) cannot be less than \(n-m+1\).

Proof: rearrange the rows of the matrix so that the first \(m-1\) rows have no linear dependence on the other rows. Then each of the other \(n-m+1\) rows of the matrix, together with the first \(m-1\) rows, gives \(n-m+1\) nonzero determinants. In the error equations (49), the number of rows having no linear dependence on the other rows is always \(\ge m\), i.e., one can always obtain more than \(n-m+1\) solutions.

Obviously,

\[ X_1=X_2=\ldots=X_{n-m+1} \tag{52} \]

Multiply \(X_1, X_2,\ldots\) by undetermined multipliers \(p_1,p_2,\ldots\), subject to the condition

\[ p_1+p_2+\ldots+p_{n-m+1}=1, \tag{53} \]

and form the sum

\[ X=X_1p_1+X_2p_2+\ldots+X_{n-m+1}p_{n-m+1} \tag{54} \]

Equation (54), if in it the values \(X_i\) from (50), (51) are substituted, will determine the true value \(X\) of the required quantity, expressed as a function of all the quantities \(L_i\) subject to measurement. Undoubtedly, equation (54) is valid only under the equalities (52) and (53). Only in this case will equation (54) give the true value of the quantity sought.

Suppose that, from measurements for the quantities \(L_i\), the values \(l_i\) have been found. Then from (50), (51), … approximate values \(x_i\) of the unknown \(X\) will be found, and equation (54) will give its approximate value \(x\), depending on the multipliers \(p_i\),

\[ x=[x_i p_i],\qquad (i=1,2,\ldots,n-m+1). \tag{55} \]

Let us find the multipliers \(p\) so that

\[ m_x^2=minimum. \tag{56} \]

Then, replacing in (50), (51), …, \(L_i\) by \(l_i\) and applying formula (1), we find:

\[ m_x^2 = F(l_i,p_i);\qquad (i=1,2\ldots n-m+1) \tag{57} \]

Equating the partial derivatives of (57) with respect to the variables \(p\) to the partial derivatives of equation (53), multiplied by \(2k\), we find the values \(p_i\), and then, from (55), \(x_0\). Equation (55) may be regarded as the arithmetic mean of dependent values \(x_i\), found from the algebraic solution of \(n-m+1\) systems of equations. From (55) and (56) it follows that \(p\) have the meaning of the weights of these values, satisfying the requirement of the greatest weight of the sought quantity. Substituting \(p\) in equation (57), we find \(m_x^2\).

It is important to note the circumstance that the described course of solving the problem is also applicable in the case when each of the equations (49) contains more than one quantity \(L\).¹

If the system (49) is solved under the condition \([v^2]=\min\), i.e., if \(x\) is found from its normal equations

\[ [aa]x+[ab]y+\cdots +[al]=0 \]

\[ [ab]x+[bb]y+\cdots +[bl]=0, \tag{58} \]

then in general form it can be shown that the expressions \(x^\circ\) from (58) and (55) coincide.

To avoid cumbersome calculations, let us apply formulas (49)—(57) to three equations with two unknowns and show that by theorem 2 and by the method of least squares we obtain one and the same result. Since the method of derivation does not depend on the number of equations, the generality of the proof for three equations is preserved.

Let the equations be:

\[ a_1X+b_1Y-L_1=0, \]

\[ a_2X+b_2Y-L_2=0, \]

\[ a_3X+b_3Y-L_3=0, \tag{59} \]

then from the solution of the first two equations we have

\[ X=\frac{1}{D_{12}}(b_2L_1-b_1L_2); \]

from the second and third equations we obtain:

\[ X=\frac{1}{D_{23}}(b_3L_2-b_2L_3); \]

then

\[ X=\frac{1}{D_{12}}(b_2L_1-b_1L_2)p_1+\frac{1}{D_{23}}(b_3L_2-b_2L_3)p_2, \tag{60} \]

where

\[ p_1+p_2=1. \tag{61} \]

By \(D_{12}\) is denoted the determinant of the system consisting of the first and second equations; by \(D_{23}\), the determinant of the system consisting of the second and third equations.

Let the equally accurate values \(l_i\), with mean square error \(m=1\), have been found from the measurements. Then, on the basis of (56) and (60),

\[ m_x^2=\frac{[b^2]_3}{D_{12}^2}p_1^2+\frac{[b^2]_1}{D_{23}^2}p_2^2-2\,\frac{b_1b_3}{D_{12}D_{23}}p_1p_2, \tag{62} \]

where \([b^2]_3\) denotes the sum of the \(b_i^2\) without \(b_3^2\); \([b^2]_1\) is the sum of the \(b_i^2\) without \(b_1^2\).

The most advantageous solution will be that which gives a value of \(x\) as close as possible to the true value (60). This is achieved if

\[ m_x^2=minimum. \]

Then, on the basis of (62) and (61), we have

\[ \frac{\partial(m_x^2)}{2\partial p_1} =\frac{[b^2]_3}{D_{12}^2}p_1-\frac{b_1b_3}{D_{12}D_{23}}p_2=k, \]

\[ \frac{\partial(m_x^2)}{2\partial p_2} =-\frac{b_1b_3}{D_{12}D_{23}}p_1+\frac{[b^2]_1}{D_{23}^2}p_2=k . \tag{63} \]

The solution of this system gives

\[ p_1=\frac{1}{D}\left(\frac{[b^2]_1}{D_{23}^2}+\frac{b_1b_3}{D_{12}D_{23}}\right)k, \]

\[ D=\frac{b_2^2[b^2]}{D_{12}^2D_{23}^2}; \tag{64} \]

\[ p_2=\frac{1}{D}\left(\frac{[b^2]_3}{D_{12}^2}+\frac{b_1b_3}{D_{12}D_{23}}\right)k, \]

where \(D\) is the determinant of the system (63).

Denoting the bracket on the right-hand side of the expression for \(p_1\) by \(A_1\), and in the expression for \(p_2\) by \(A_2\), and substituting \(p_1\) and \(p_2\) into equation (61), we shall have

\[ k=\frac{D}{A_1+A_2}, \]

or

\[ p_1=\frac{A_1}{A_1+A_2};\qquad p_2=\frac{A_2}{A_1+A_2}. \tag{65} \]

Thus the multipliers \(p_1\) and \(p_2\) have been found. The most advantageous value of \(x\) from (60) is determined as follows:

\[ x=\frac{1}{D_{12}}(b_2l_1-b_1l_2)p_1+\frac{1}{D_{23}}(b_3l_2-b_2l_3)p_2 . \tag{66} \]

Let us solve system (59) by the method of indirect observations, replacing \(L_i\) in it by \(l_i\). Forming the corresponding system of normal equations,

\[ [aa]x+[ab]y-[al]=0 \]

\[ [ab]x+[bb]y-[bl]=0, \tag{67} \]

we obtain

\[ x=\frac{1}{D}([bb][al]-[ab][bl]); \]

or, more fully,

\[ x=\frac{1}{D}\left\{(a_1[bb]-b_1[ab])l_1+(a_2[bb]-b_2[ab])l_2\cdots\right\}, \tag{68} \]

where \(D\) is the determinant of the system (67).

If we show that the coefficients of \(l_i\) in expressions (66) and (68) coincide, then this will prove that the requirement \([v^2]=min\) and Theorem 2 lead to one and the same result of adjustment.

Turning to equation (66), we see that the coefficient of \(l_1\), on the basis of (65) and (64), is equal to

\[ \sigma \frac{b_2}{D_{12}}p_1 = b_2 \frac{[b^2]_1D_{12}+b_1b_3D_{23}} {[b^2]_1D_{12}^2+[b^2]_3D_{23}^2+2b_1b_3D_{12}D_{23}}. \]

Substituting the values of the determinants and \([b^2]_1,\ldots\), carrying out the multiplication and taking \(a_1\) and \(b_1\) outside the parentheses, we find that the numerator is equal to \((a_1[bb]-b_1[ab])b_2\). Applying the same operations to the denominator, we find that it is equal to

\[ b_2^2(b_2^2a_1^2+a_2^2b_1^2-2a_1b_1a_2b_2+a_2^2b_3^2-2a_1b_1a_3b_3+a_3^2b_1^2+a_2^2b_3^2+b_1^2a_3^2- \]

\[ -2a_2b_2a_3b_3). \]

Add to the expression standing in parentheses the identity

\[ 0=[a^2b^2]-[a^2b^2]; \]

then the denominator can be brought to the form \(b_2^2([aa][bb]-[ab]^2)=Db_2^2\), whence it follows that the coefficient of \(l_1\) in equation (66)

\[ \frac{b_2}{D_{12}}p_1 = \frac{a_1[bb]-b_1[ab]}{D} \]

coincides with the coefficient of \(l_1\) in equation (68). In exactly the same way one can show that the coefficients of \(l_2\) and \(l_3\) in equations (66) and (68) are equal, i.e. the adjusted values of the unknown under the condition \([v^2]=min\) and by Theorem 2 are also equal.

Consider an example consisting of three equations with two unknowns:

\[ \begin{aligned} 2X-Y-L_1&=0,\\ X+Y-L_2&=0,\\ X-Y-L_3&=0. \end{aligned} \tag{a} \]

Let the measured values \(l_i\) be obtained as

\[ l_1=4,\qquad l_2=5,\qquad l_3=2; \]

the corresponding normal equations will be:

\[ 6x-2y-15=0 \]

\[ -2x+3y+1=0, \]

whence

\[ x=3.072,\qquad m_x^2=0.214. \]

Such is the result obtained from the formulas of the method of least squares.

Let us solve system (a) by formulas (60)—(66), for which we compute the determinants \(D_{ik}\) and their squares

	\(D_{12}\)	\(D_{13}\)	\(D_{23}\)
\(D_i\)	3	\(-1\)	\(-2\)
\(D_i^2\)	9	1	4

Then from the solution of the first and second equations we obtain

\[ x_{12}=\frac{1}{D_{12}}(b_2l_1-b_1l_2)=3.0. \]

The solution of the second and third equations gives a value equal to

\[ x_{23}=\frac{1}{D_{23}}(b_3l_2-b_2l_3)=3.5. \]

The adjusted value will be equal to

\[ x^0=x_{12}p_1+x_{23}p_2. \tag{b} \]

Let us compute \(p_1\) and \(p_2\), by which the results obtained must be multiplied. Using formulas (64) and (65), we find

\[ A_1=\frac{[b^2]_1}{D_{23}^{\,2}}+\frac{b_1b_3}{D_{12}D_{23}}=\frac{1}{3}, \]

\[ A_2=\frac{[b^2]_3}{D_{12}^{\,2}}+\frac{b_1b_3}{D_{12}D_{23}}=\frac{1}{18}, \tag{c} \]

\[ p_1=\frac{A_1}{A_1+A_2}=\frac{6}{7};\qquad p_2=\frac{A_2}{A_1+A_2}=\frac{1}{7}, \]

and the adjusted value \(x^0\) on the basis of (66) is computed as follows:

\[ x^0=3\cdot\frac{6}{7}+\frac{7}{2}\cdot\frac{1}{7}=3.072 \]

Substituting the values \(p_i\) into equation (62), we obtain

\[ m_{x^0}^{2}=\frac{2}{9}\cdot0.734+\frac{1}{2}\cdot0.020+\frac{1}{3}\cdot0.122=0.214, \]

i.e. we have obtained the results previously found by us by the method of least squares.

In deriving formulas (60)—(66) we did not restrict the choice of the values \(x_{ik}\), and therefore we should obtain this same result by choosing another system of values \(x\). Let us confirm what has been said by solving this same example. Take the values:

\[ x_{13}=\frac{1}{D_{13}}(b_3l_1-b_1l_3)=2.0, \]

\[ x_{23}=\frac{1}{D_{23}}(b_3l_2-b_2l_3)=3.5, \]

\[ x^0=x_{13}p_1+x_{23}p_2, \tag{b}' \]

\[ x^0=\frac{b_3p_1}{D_{13}}\,l_1+\frac{b_3p_2}{D_{23}}\,l_2-\left(\frac{b_1p_1}{D_{13}}+\frac{b_2p_2}{D_{23}}\right)l_3, \]

\[ m_{x^0}^{2}=\frac{[b^2]_2}{D_{13}^{\,2}}p_1^{2} +\frac{[b^2]_1}{D_{23}^{\,2}}p_2^{2} +2\frac{b_1b_2}{D_{13}D_{23}}p_1p_2, \tag{d} \]

\[ \frac{[b^2]_2}{D_{13}^{\,2}}p_1+\frac{b_1b_2}{D_{13}D_{23}}p_2=k, \]

\[ \frac{b_1b_2}{D_{13}D_{23}}p_1+\frac{[b^2]_1}{D_{23}^{\,2}}p_1=k. \]

\[ p_1=\frac{1}{D}\left(\frac{[b^2]_2}{D_{13}^2}-\frac{b_1b_2}{D_{13}D_{23}}\right)k, \]

\[ p_2=\frac{1}{D}\left(\frac{[b^2]_2}{D_{13}^2}-\frac{b_1b_2}{D_{13}D_{23}}\right)k. \]

The expressions standing in parentheses are \(A_1\) and \(A_2\) in equations \((c)\), or

\[ A_1=1;\qquad A_2=2.5. \]

\[ p_1=0.286;\qquad p_2=0.714 \]

The adjusted value \(x^0\) is obtained on the basis of \((b)'\)

\[ x^0=2\cdot 0.286+3.5\cdot 0.714=3.071;\qquad m_{x^0}^{2}=0.215, \]

i.e. the result of adjustment does not depend on the choice of the system of values \(x_{ik}\), as also followed from the theory.

Now let us take (following Jacobi) all combinations

\[ X_{12}=\frac{1}{D_{12}}(b_2L_1-b_1L_2);\qquad X_{13}=\frac{1}{D_{13}}(b_3L_1-b_1L_3);\qquad X_{23}=(b_3L_2-b_2L_3) \tag{e} \]

Then the true value \(X\) may be represented in the form

\[ X=X_{12}p_1+X_{13}p_2+X_{23}p_3 \tag{f} \]

under the condition

\[ [p]=1. \tag{g} \]

If in \((e)\) and \((f)\) we replace \(L_i\) by \(l_i\), then we obtain an approximate value \(x\), equal to

\[ x=\left(\frac{b_2}{D_{12}}p_1+\frac{b_3}{D_{13}}p_2\right)l_1+ \left(\frac{b_3}{D_{23}}p_3-\frac{b_1}{D_{12}}p_1\right)l_2 -\left(\frac{b_1}{D_{13}}p_2+\frac{b_2}{D_{23}}p_3\right)l_3 \tag{h} \]

with mean error \(m_x\), depending on \(p\). For \(m_l=1\)

\[ m_x^2=\frac{[b^2]_3}{D_{12}^2}p_1^2+\frac{[b^2]_2}{D_{13}^2}p_2^2+\frac{[b^2]_1}{D_{23}^2}p_3^2 +2\frac{b_2b_3}{D_{12}D_{13}}p_1p_2+ \]

\[ +2\frac{b_1b_2}{D_{13}D_{23}}p_2p_3 -2\frac{b_1b_3}{D_{12}D_{23}}p_1p_3; \tag{i} \]

Differentiating \((i)\) with respect to the variables \(p_i\), reducing by \(2\), and equating to the differentials of \((g)\) with respect to the same variables, multiplied by \(k\), we obtain:

\[ \frac{[b^2]_3}{D_{12}^2}p_1+ \frac{b_2b_3}{D_{12}D_{13}}p_2 -\frac{b_1b_3}{D_{12}D_{23}}p_3=k; \]

\[ \frac{b_2b_3}{D_{12}D_{13}}p_1+ \frac{[b^2]_2}{D_{13}^2}p_2+ \frac{b_1b_2}{D_{13}D_{23}}p_3=k; \tag{j} \]

\[ -\frac{b_1b_3}{D_{12}D_{23}}p_1+ \frac{b_1b_2}{D_{13}D_{23}}p_2+ \frac{[b^2]_1}{D_{23}^2}p_3=k. \]

Let \(p_1\) from equations \((j)\) be equal to

\[ p_1=\frac{A}{D_0}. \]

Let us show that \(D_0=0\)

\[ D_0=-\frac{1}{D_{12}^{2}D_{13}^{2}D_{23}^{2}} \begin{vmatrix} [b^2]_3 & b_2b_3 & -b_1b_3\\ b_2b_3 & [b^2]_2 & b_1b_2\\ -b_1b_3 & b_1b_2 & [b^2]_1 \end{vmatrix} \]

Expanding the determinant, we obtain:

\[ \mathrm{I}=\left\{([b^2]-b_2^2)([b^2]-b_1^2)-b_1^2b_2^2\right\}[b^2]_3; \]

\[ \mathrm{II}=-b_2b_3\left\{b_2b_3([b^2]-b_1^2)+b_1^2b_2b_3\right\} =-b_2^2b_3^2[b^2]; \]

\[ \mathrm{III}=-b_1^2b_3\left\{b_1b_3([b^2]-b_2^2)+b_0b_2^2b_3\right\} =-b_2^2b_3^2[b^2]; \]

\[ D_0=\mathrm{I}+\mathrm{II}+\mathrm{III} =(b_1^2+b_2^2)[b^2]\,b_3^2-b_2^2b_3^2[b^2]-b_1^2b_3^2[b^2]=0. \]

It can also be shown that the determinant of the system \((j)\) will be equal to zero also in the case when we take not all the combinations, but only \(n-m+2\).

For this purpose let us use the example of Gauss, given on p. 233 of Whittaker and Robinson’s work The Mathematical Treatment of Observational Data.

\[ x-y+2z=3 \]

\[ 3x+2y-5z=5 \]

\[ 4x+y+4z=2; \]

\[ -x+3y+3z=14. \]

First let us solve the required number of combinations of equations:

\[ n-m+1=2,\qquad \text{since } n=4,\quad m=3. \]

Take the combinations \(x_{123},\ x_{234}\)

\[ \left. \begin{aligned} x_{123} &=\frac{1}{D'_0}\left\{(b_2c_3-b_3c_2)l_1-(b_1c_3-b_3c_1)l_2+(b_1c_2-b_2c_1)l_3\right\} \\ &=\frac{1}{D'_0}(D_1l_1-D_2l_2+D_3l_3), \\[1em] x_{234} &=\frac{1}{D''_0}\left\{(b_3c_4-b_4c_3)l_2-(b_2c_4-b_4c_2)l_3+(b_2c_3-b_3c_2)l_4\right\} \\ &=\frac{1}{D''_0}(D_4l_2-D_5l_3+D_6l_4), \\[1em] x^0&=x_{123}p_1+x_{234}p_2;\qquad [p]=1. \end{aligned} \right\} \tag{l} \]

The mean square error of \(x^0\), on the basis of \((l)\), is equal to

\[ m_{x^0}^{2} = \frac{1}{D_0^{\prime\,2}}(D_1^2+D_2^2+D_3^2)p_1^2 + \frac{1}{D_0^{\prime\prime\,2}}(D_4^2+D_5^2+D_6^2)p_2^2 - \frac{2}{D'_0D''_0}(D_2D_4+D_3D_5) \tag{m} \]

Then

\[ -\frac{1}{D_0'^2}(D_1^2+D_2^2+D_3^2)p_1-\frac{1}{D_0'D_0''}(D_2D_4+D_3D_5)p_2=k \]

\[ -\frac{1}{D_0'D_0''}(D_2D_4+D_3D_5)p_1+\frac{1}{D_0''^2}(D_4^2+D_5^2+D_6^2)p_2=k. \tag{n} \]

Let us compute \(D_i\)

	\(D_1\)	\(D_2\)	\(D_3\)	\(D_4\)	\(D_5\)	\(D_6\)
\(D_i\)	13	−6	1	−9	21	13
\(D_i^2\)	169	36	1	81	441	169

Equations \((n)\) give

\[ p_1=0{,}15492,\qquad p_2=0{,}84508. \]

Since (according to Wittaker)

\[ x_{123}=\frac{90}{35}=2{,}571428,\qquad x_{234}=\frac{304}{124}=2{,}451612, \]

then

\[ x^0=x_{123}p_1+x_{231}p_2=2{,}470174, \tag{p} \]

i.e. we have obtained the same result as by Jacobi’s theorem.

In formula \((p)\) only two solutions are used; the weights \(p\) are subject to the condition of greatest weight of the sought quantity. We shall show that the introduction of even one solution in excess of \(n-m+1\) reduces the determinant of the system \((n)\) to zero.

Let us add one more to the two necessary solutions; suppose that

\[ x_{121}=\frac{1}{D_0'''}\{(b_2c_1-b_1c_2)l_1-(b_1c_1-b_1c_1)l_2+(b_1c_2-b_2c_1)l_4\}= \]

\[ =\frac{1}{D_0'''}(D_7l_1-D_8l_2+D_9l_4) \tag{q} \]

Then the true value \(X\) will be equal to

\[ X=X_{123}p_1+X_{231}p_2+X_{121}p_3,\qquad \text{where } [p]=1. \]

The approximate value \(x\) on the basis of \((i)\) and \((q)\) is found from the expression

\[ x=\left(\frac{D_1}{D_0'}p_1+\frac{D_7}{D_0'''}p_3\right)l_1+ \left(-\frac{D_2}{D_0'}p_1+\frac{D_4}{D_0''}p_2-\frac{D_8}{D_0'''}p_3\right)l_2+ \]

\[ +\left(\frac{D_3}{D_0'}p_1-\frac{D_5}{D_0''}p_2\right)l_3+ \left(\frac{D_6}{D_0''}p_2+\frac{D_9}{D_0'''}p_3\right)l_4 \]

with mean error

\[ m_x^2=\frac{1}{D_0'^2}(D_1^2+D_2^2+D_3^2)p_1^2+\frac{1}{D_0''^2}(D_4^2+D_5^2+D_6^2)p_2^2+ \]

\[ +\frac{1}{D_0'''^2}(D_7^2+D_8^2+D_9^2)p_3^2-\frac{2}{D_0'D_0''}(D_2D_4+D_3D_5)p_1p_2+ \]

\[ + \frac{2}{D'_0 D'''_0}(D_1D_7 + D_2D_8)p_1p_3 + \frac{2}{D''_0 D'''_0}(D_6D_9 - D_4D_8)p_2p_3 = \]

\[ = \frac{1}{D_0'^2}Ap_1^2 + \frac{1}{D_0''^2}Bp_2^2 + \frac{1}{D_0'''^2}Cp_3^2 - \frac{2D}{D'_0D''_0}p_1p_2 \]
\[ + \frac{2E}{D'_0D_0''^2}p_1p_3 + \frac{2F}{D''_0D'''_0}p_2p_3 . \]

Equating the partial derivatives of \(m_x^2\) and \(2[p]k\) with respect to the variables \(p_i\), we obtain:

\[ \frac{A}{D_0'^2}p_1 - \frac{D}{D'_0D''_0}p_2 + \frac{E}{D'_0D_0'''^2}p_3 = k \]

\[ -\frac{D}{D'_0D''_0}p_1 + \frac{B}{D_0''^2}p_2 + \frac{F}{D''_0D'''_0}p_3 = k \tag{r} \]

\[ \frac{E}{D'_0D'''_0}p_1 + \frac{F}{D''_0D'''_0}p_2 + \frac{C}{D_0'''^2}p_3 = k \]

The determinant of system \((r)\) will also be equal to zero. Omitting the cumbersome expansion of the determinant in the general form, we shall show, on the same Gauss example, that \(D_0 = 0\).

\[ D_0 = \frac{1}{D_0'^2D_0''^2D_0'''^2} \begin{vmatrix} A & -D & E\\ -D & B & F\\ E & F & C \end{vmatrix} \]

It is necessary to show that the expression

\[ A(BC - F^2) - D(DC - EF) - E(DF + BE); \]

is equal to zero on the basis of \((k)\) and \((q)\), where the values of the determinants \(D_i\) are given by the table

	\(D_7\)	\(D_8\)	\(D_9\)
\(D_i\)	21	\(-9\)	1
\(D_i^2\)	441	81	1

Then, also taking \((r)\) into account, we obtain:

\[ A = 206,\quad B = 691,\quad C = 523,\quad D = 75,\quad E = 327,\quad F = -68, \]
\[ D_0 = 206(361393 - 4624) + 75(22236 - 39225) + 327(510 - 225957) = +73494414 - 73494414 = 0. \]

Above we found the weights \(p_1\) and \(p_2\), satisfying the condition of the greatest weight \(x^0\). From equation \((p)\) it follows that the number of combinations required is \(n - m + 1 = 2\); according to Jacobi, however, there should be four of them.

We cannot regard the quantities \(p_1\) and \(p_2\) otherwise than as weights with which the dependent values \(x_{12}\) and \(x_{234}\) enter into the arithmetic mean \((p)\). In this case \(P_x^0 \ne p_1 + p_2\). Using \(p_1\) and \(p_2\), we shall find the value of \(m_x^{2\,0}\), i.e. also \(p_x^0\).

Substituting the values of \(D\) and \(p\) into \((i)\), we obtain:

\[ m_x^{2\,0} = \frac{206}{1225}p_1^2 + \frac{69}{15376}p_2^2 + 2\frac{75}{4340}p_1p_2 = 0{,}0406, \]

which is in complete agreement with the data of Gauss’s example, given by Wittaker (p. 234).

Conclusions

The results of the investigation may be formulated as follows.

1. The method of least squares is defined for a certain mean of a series of observations; the given series of observations is used by it independently of the distribution of the error values in this series.

2. Theorem 1. The most probable value of each of \(n\) observed quantities connected by \(r\) exact conditions \((n>r)\) can be represented in the form of the arithmetic mean of the result of a direct measurement of the sought quantity and of its dependent values obtained from \(r\) simultaneous solutions of the equations, or supplied by each conditional equation, with weights satisfying the requirement of the greatest weight of the sought quantity.

3. Theorem 2. The most probable value of each of \(m\) functions of direct observations connected with the direct observations by \(n\) equations \((n>m)\) can be represented in the form of the arithmetic mean of the dependent values obtained from the solution of the necessary and sufficient \((n-m+1)\) number of combinations of equations, with weights satisfying the requirement of the greatest weight of the sought quantity.

4. These theorems have theoretical and practical consequences and extend the domain of application of the method of least squares.

5. The weights with which the values of the sought quantity enter into the arithmetic mean make it possible to calculate also the weight of its adjusted value.

6. The arithmetic mean of independent values is a special case of the principle, underlying the method of least squares, of the greatest weight of the sought quantity.

1. The method described is also applicable in the most general case, when equations (49) have the form
   \[ 0=-A_iL_1-B_iL_2-\cdots +a_iX+b_iY+c_iZ+\cdots . \] ↩↩↩

2. The number of equations and unknowns does not affect the method of derivation, i.e. the choice of two equations does not diminish the generality of the proof. ↩

3. With unequal observations, the essence of the question does not change. ↩