Translation:Supplementum theoriae combinationis observationum erroribus minimis obnoxiae

1.
In the previous Memoir, we assumed that the quantities to be determined, with the help of imperfect observations, depended on certain unknown elements, in terms of which they could be expressed: the problem then consisted in deducing from the observations, as accurately as possible, the values of these elements.

In most cases, this is indeed how the question arises. But sometimes the problem is presented differently, and at first glance it is even doubtful whether or not it can be reduced to a problem of the required form. Indeed, frequently the quantities under observation are not given in the form of a function of certain elements, and they seem to only be reducible to such a form by difficult or even ambiguous operations; whereas the nature of the problem does provide certain conditions that the observed values must rigorously satisfy.

Upon closer inspection, however, it can be seen that this case does not differ essentially from the previous one, and that it can be reduced to it. Indeed, if $$\pi$$ denotes the number of observed quantities, and $$\sigma$$ denotes the number of constraint equations, then nothing prevents us from choosing, among the former, $$\pi - \sigma$$ quantities, and considering them as our only unknowns, the others being regarded as functions of these, defined by the constraint equations. By this artifice, we can return to the case of the previous Memoir.

Nevertheless, although this approach often leads to the result in a quite convenient manner, one cannot deny that it is less natural, and it is therefore desirable to treat the problem in another form, which admits a very elegant solution. Moreover, since this new solution leads to faster calculations than the previous one whenever $$\sigma$$ is less than $$\tfrac{1}{2}\,\pi,$$ or equivalently, whenever the multitude of elements that we have denoted by $$\rho$$ in the previous commentary, is greater than $$\tfrac{\pi}{2},$$ it is preferable to use the new solution, even if it would be easy, given the nature of the problem, to unambiguously eliminate the constraint equations.

2.
Let $$v,$$ $$v',$$ $v'',$ etc. denote the $$\pi$$ quantities whose values are provided by observation. Suppose an unknown depends on these and is expressed by a known function $$u;$$ let $$l,$$ $$l',$$ $l'',$ etc. be the values of the differential quotients

corresponding to the true values of $$v,$$ $$v',$$ $v'',$ etc.. If one substituted the true values for $$v,$$ $$v',$$ $v,$ etc. in the function $$u,$$ one would also obtain the true value of $$u;$$ but if the observations are affected by errors $$e,$$ $$e',$$ $e,$ etc., there resulting total error for $$u$$ will be represented by

provided (when $$u$$ is not linear) that one can neglect the squares and products of errors $$e,$$ $$e',$$ $$e'',$$ etc., as we will always assume.

Although the magnitude of errors $$e,$$ $$e',$$ $e,$ etc., is uncertain, the uncertainty attached to the value found for $$u$$ can generally be measured by the mean error to be feared'' in the adopted determination. According to the principles developed in the previous commentary, this average error is

where $$m,$$ $$m',$$ $m'',$ etc. are the average errors of the various observations. If all observations are affected by the same degree of uncertainty, this expression becomes

and it is clear that in this calculation, it is permissible to replace $$l,$$ $$l',$$ $l'',$ etc., with the values taken by the differential coefficients

when one replaces $$v,$$ $$v',$$ $v'',$ etc. with their observed values.

3.
When the quantities $$v,$$ $$v',$$ $v'',$ etc. are independent, the unknown can only be determined in one way, and the uncertainty attached to the result cannot be avoided or reduced. In this case, the observations provide a value of the unknown which is not arbitrary.

It is quite different when the quantities $$v,$$ $$v',$$ $v'',$ etc. are subject to certain relations, which we will suppose are expressed by $$\sigma$$ conditional equations

where $${X},$$ $${Y},$$ ${Z},$ etc. are given functions of the unknowns $$v,$$ $$v',$$ $v,$ etc. In this case, there are infinitely many ways to represent our unknown as a combination of the quantities $$v,$$ $$v',$$ $v,$ etc., since it is clear that the function $$u$$ can be replaced by any other function $${U}$$ such that the difference $${U} - u$$ vanishes identically upon setting

If the observations were rigorously exact, this substitution would not change the result at all; but, due to inevitable errors, each form adopted for $$u$$ will correspond to a different result, and the error committed, instead of being

will be

where $${L},$$ $${L}',$$ ${L}'',$ etc. denote the differential quotients

Although it is impossible to assign the value of the various errors, we can, however, compare the mean errors to be feared in the various combinations. The most advantageous combination will be the one that gives the minimum value to the mean error. This error being

we must seek to make the sum

as small as possible.

4.
The infinite number of functions $${U},$$ by which $$u$$ can be replaced, will differ from each other in our investigations only by the values they provide for $${L},$$ $${L}',$$ ${L}'',$ etc.: therefore, we must first seek the relationships that exist between the systems of values that these coefficients can take. Let

denote the values taken by the coefficients

when we substitute the true values of $$v,$$ $$v',$$ $v,$ etc. It is clear that if increments $${d} v,$$ $${d} v',$$ ${d} v,$ etc. are given to $$v,$$ $$v',$$ $v'',$ etc. which do not change $${X},$$ $${Y},$$ ${Z},$ etc. and thus leave each of them with a value of zero, these increments, which satisfy the equations

will not change the value of $${U}-u,$$ and consequently, we will have

We easily conclude that $${L},$$ $${L}',$$ ${L}'',$ etc., must have the form

where $$x,$$ $$y,$$ $z,$ etc. are determined multipliers. Conversely, it is clear that for all values of $$x,$$ $$y,$$ $z,$ etc., we can form a function $${U},$$ for which the values $${L},$$ $${L'},$$ ${L''},$ etc., will precisely be those provided by these equations, and this function can, according to the above, be substituted for $$u.$$ The simplest form we can give it is

but the most general is

where $$u'$$ denotes a function of $$v,$$ $$v',$$ $v'',$ etc. which vanishes identically when $${X},$$ $${Y},$$ ${Z},$ etc., are zero, and whose value, in the current case, will be maximum or minimum. However, this makes no difference for our purposes.

5.
It is now easy to assign values to $$x,$$ $$y,$$ $z,$ etc. such that the sum

achieves its minimum value. It is clear that to achieve this goal, the knowledge of the absolute mean errors $$m,$$ $$m',$$ $m'',$ etc. is not necessary. Rather, it suffices to know only their ratios. Indeed, instead of these quantities, let us introduce the weights of the observations, $$p,$$ $$p',$$ $p,$ etc., i.e. numbers inversely proportional to the squares $$m^2,$$ $${m'}^2,$$ ${m}^2,$ etc. The quantities $$x,$$ $$y,$$ $z,$ etc. must then be determined in such a way that the polynomial

achieves its minimum value. Suppose that $$x^0,$$ $$y^0,$$ $z^0,$ etc., are the determined values of $$x,$$ $$y,$$ $z,$ etc. that correspond to this minimum.

We now introduce the following notation:

The minimality condition clearly requires that we have {{MathForm2|(1)|$$\left.\begin{array}{l}\begin{alignat}{4} 0 {}={} &(aa)\,&x^0 {}+{} &(ab)\,&y^0 {}+{} &(ac)\,&z^0 + \ldots {}+{} &(al), \\ 0 {}={} &(ba)\,&x^0 {}+{} &(bb)\,&y^0 {}+{} &(bc)\,&z^0 + \ldots {}+{} &(bl), \\ 0 {}={} &(ca)\,&x^0 {}+{} &(cb)\,&y^0 {}+{} &(cc)\,&z^0 + \ldots {}+{} &(cl), \end{alignat} \\ \;\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots \end{array}\right\}\quad $$}}

After these equations have provided the values of $$x^0,$$ $$y^0,$$ $z^0,$ etc., we set

{{MathForm2|(2)|$$\left.\begin{array}{l}\begin{alignat}{5} &a &x^0 {}+{} &b  &y^0 {}+{} &c  &z^0 + \ldots {}+{} &l  &{}={}& {L}, \\ &a' &x^0 {}+{} &b' &y^0 {}+{} &c' &z^0 + \ldots {}+{} &l' &{}={}& {L}', \\ &a&x^0 {}+{} &b&y^0 {}+{} &c&z^0 + \ldots {}+{} &l&{}={}& {L}'', \end{alignat} \\ \,\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdot\cdot \end{array} \right\}\quad $$}} and the most suitable function to determine our unknown, corresponding to the minimum mean error, will be one whose differential coefficients, for the considered values of the variables, are equal to $${L},$$ $${L'},$$ ${L''},$ etc. The weight of the determination thus obtained will be

meaning that $$\frac{1}$$ will precisely be the value taken by the polynomial considered above, for the values of the variables $$x,$$ $$y,$$ $z,$ etc. which satisfy the equations (1).

6.
In the previous article, we showed how to determine the function $${U}$$ that provides the most suitable determination of the unknown $$u.$$ Let us now examine what value results from this. Let's denote it by $${K}$$: it will be obtained by substituting, in $${U},$$ the observed values of the quantities $$v,$$ $$v',$$ $v,$ etc. Let $$k$$ be the value that $$u$$ takes when the same substitutions are made, and let $$\varkappa$$ be the true value of the unknown, obtained by substituting the true values of $$v,$$ $$v',$$ $v,$ etc., either in $$u,$$ or in $${U}.$$ We have

and, consequently,

Replacing $${L}-l,$$ $${L'}-l',$$ ${L}-l,$ etc. with their values provided by (2), and setting {{MathForm2|(4)|$$\left\{\begin{array}{l}\begin{alignat}{4} &a\, &e {}+{} &a' &e' {}+{} &a &e + \ldots &{}={} \mathfrak{A}, \\ &b\, &e {}+{} &b' &e' {}+{} &b &e + \ldots &{}={} \mathfrak{B}, \\ &c\, &e {}+{} &c' &e' {}+{} &c &e + \ldots &{}={} \mathfrak{C}, \end{alignat} \\ \;\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdot\cdot \end{array}\right.$$}} we will have

It is not possible to calculate $$\mathfrak{A},$$ $$\mathfrak{B},$$ $\mathfrak{C},$ etc. by means of the formulas (4), as the errors $$e,$$ $$e',$$ $e,$ etc. appearing in them have unknown values, but it is easy to see that the quantities $$\mathfrak{A},$$ $$\mathfrak{B},$$ $\mathfrak{C},$ etc., are none other than the values of $${X},$$ $${Y},$$ ${Z},$ etc., corresponding to the observed values of $$v,$$ $$v',$$ $v,$ etc. In this way, the systems of equations (1), (3), (5) exhibit the complete solution to our problem. In fact, it is clear that the remark made at the end of art. 2, regarding the quantities $$l,$$ $$l',$$ $l,$ etc., can be applied to the calculation of $$a,$$ $$a',$$ $a,$ etc. $$b,$$ $$b',$$ $b,$ etc. etc., or in other words, the true values of $$v,$$ $$v',$$ $v,$ etc. can be replaced with the observed values.

7.
In place of formula (3), which represents the weight of the most probable determination, several other expressions can be given, which are worth developing.

First, observe that if the equations (2) are multiplied by $$\frac{a}{p},$$ $$\frac{a'}{p'},$$ $\frac{a}{p},$ etc. resp. and then added, the result is

The left hand side is $$=0,$$ and therefore, denoting the right hand side by $$(a\,{L}),$$ we have

and similarly

Next, multiplying the equations (2) by $$\frac{p},$$ $$\frac{{L}'}{p'},$$ $\frac{p},$ etc.|undefined and adding, we find that

and thus we obtain a second expression for the weight,

If, finally, if we multiply the same equations (2) by $$\frac{l}{p},$$ $$\frac{l'}{p'},$$ $\frac{l}{p},$ etc. and add, we obtain a third expression for the weight,

where, with the same notation as before,

One can easily derive from this a fourth expression for the weight,

8.
The general solution we have just outlined is particularly adapted to the case where there is only one unknown to determine. However, when seeking the most plausible values for several unknowns, all of which depend on the same observations, or when it is unknown which unknowns should be derived from the observations, it is necessary to proceed differently, as we will now explain.

Let us consider the quantities $$x,$$ $$y,$$ $z,$ etc. as indeterminate, and set {{MathForm2|(6)|$$\left.\begin{array}{l}\begin{alignat}{4} &(aa)\, &x {}+{} &(ab)\, &y {}+{} &(ac)\, &z + \ldots &{}={} \xi, \\ &(ab)\, &x {}+{} &(bb)\, &y {}+{} &(bc)\, &z + \ldots &{}={} \eta, \\ &(ac)\, &x {}+{} &(bc)\, &y {}+{} &(cc)\, &z + \ldots &{}={} \zeta, \end{alignat} \\ \;\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdot\cdot \end{array} \right\}$$}} Suppose that we deduce, by elimination, that {{MathForm2|(7)|$$\left.\begin{array}{l}\begin{alignat}{4} &(\alpha\alpha)\, &\xi {}+{} &(\alpha\beta)\, &\eta {}+{} &(\alpha\gamma)\, &\zeta + \ldots &{}={} x, \\ &(\beta \alpha)\, &\xi {}+{} &(\beta \beta)\, &\eta {}+{} &(\beta \gamma)\, &\zeta + \ldots &{}={} y, \\ &(\gamma\alpha)\, &\xi {}+{} &(\gamma\beta)\, &\eta {}+{} &(\gamma\gamma)\, &\zeta + \ldots &{}={} z, \end{alignat} \\ \;\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots \end{array}\right\}$$}}

Let us first of all observe that the symmetrically placed coefficients are necessarily equal, i.e.

This follows from the theory of elimination, but we will also give a direct demonstration below.

Thus we have {{MathForm2|(8)|$$\left.\begin{array}{l}\begin{alignat}{4} x^0 {}={} & {}-{} (\alpha\alpha)\,&(al) {}-{} &(\alpha\beta)\,&(bl) {}-{} &(\alpha\gamma)\,&(cl) - \ldots, \\ y^0 {}={} & {}-{} (\beta\alpha)\,&(al) {}-{} &(\beta\beta)\,&(bl) {}-{} &(\beta\gamma)\,&(cl) - \ldots, \\ z^0 {}={} & {}-{} (\gamma\alpha)\,&(al) {}-{} &(\gamma\beta)\,&(bl) {}-{} &(\gamma\gamma)\,&(cl) - \ldots, \end{alignat} \\ \;\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots \end{array}\right\}$$}} and so, by setting {{MathForm2|(9)|$$\left. \begin{array}{l}\begin{alignat}{4} &(\alpha\alpha)\, &\mathfrak{A} {}+{} &(\alpha\beta)\, &\mathfrak{B} {}+{} &(\alpha\gamma)\, &\mathfrak{C} + \ldots &{}={} {A}, \\ &(\beta \alpha)\, &\mathfrak{A} {}+{} &(\beta \beta)\, &\mathfrak{B} {}+{} &(\beta \gamma)\, &\mathfrak{C} + \ldots &{}={} {B}, \\ &(\gamma\alpha)\, &\mathfrak{A} {}+{} &(\gamma\beta)\, &\mathfrak{B} {}+{} &(\gamma\gamma)\, &\mathfrak{C} + \ldots &{}={} {C}, \end{alignat} \\ \;\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdot\cdot \end{array}\right\}$$}} we obtain

and if we further set {{MathForm2|(10)|$$\left. \begin{array}{l}\begin{alignat}{5} &a &{A} {}+{} &b  &{B} {}+{} &c  &{C} + \ldots &{}={} p  && \varepsilon, \\ &a' &{A} {}+{} &b' &{B} {}+{} &c' &{C} + \ldots &{}={} p' && \varepsilon', \\ &a&{A} {}+{} &b&{B} {}+{} &c&{C} + \ldots &{}={} p&&\varepsilon'', \end{alignat} \\ \;\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots \end{array}\right\}$$}} it follows that

9.
Comparing equations (7) and (9) shows that the auxiliary quantities $${A},$$ $${B},$$ ${C},$ etc., are the values taken by the indeterminates $$x,$$ $$y,$$ $z,$ etc., when we set $$\xi = \mathfrak{A},$$ $$\eta = \mathfrak{B},$$ $\zeta = \mathfrak{C}$ etc., and thus we have {{MathForm2|(12)|$$\left.\begin{array}{l}\begin{alignat}{4} &(aa)\, &{A} {}+{} &(ab)\, &{B} {}+{} &(ac)\, &{C} + \ldots &{}={} \mathfrak{A}, \\ &(ba)\, &{A} {}+{} &(bb)\, &{B} {}+{} &(bc)\, &{C} + \ldots &{}={} \mathfrak{B}, \\ &(ca)\, &{A} {}+{} &(cb)\, &{B} {}+{} &(cc)\, &{C} + \ldots &{}={} \mathfrak{C}, \end{alignat} \\ \;\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdot \end{array}\right\}$$}} Multiplying equations (10) by $$\frac{a}{p},$$ $$\frac{a'}{p'},$$ $\frac{a}{p},$ etc. respectively, and adding, we obtain } Now, since $$\mathfrak{A}$$ is the value taken by $${X}$$ after the observed values are substituted for $$v,$$ $$v',$$ $v,$ etc., it is easy to see that if we apply the corrections $$-\varepsilon,$$ $$-\varepsilon',$$ $-\varepsilon,$ etc., to each of these quantities, the value of $${X}$$ becomes $$0$$, and similarly $${Y},$$ ${Z},$ etc., vanish under this assumption. In a similar way, equation (11) also shows that $${K}$$ is the value taken by $$u$$ after the same substitutions have been made.

We will refer to the application of the corrections $$-\varepsilon,$$ $$-\varepsilon',$$ $-\varepsilon,$ etc. as compensation of observations.'' This evidently leads us to the following very important consequence. Namely, the observations, compensated in the way we have describde, exactly satisfy all of the conditional equations, and cause any function of the observed quantities to take the value resulting from the most suitable combination of the unaltered observations. Since the conditional equations are too few to deduce the exact values of the errors, we will have found, by the above, the most plausible errors. We will henceforth refer to the quantities $$\varepsilon,$$ $$\varepsilon',$$ $\varepsilon'',$ etc. by this name.

10.
Since the number of observations is greater than the number of conditional equations, an infinite number of systems can be found, other than the system of the most plausible corrections, which exactly satisfy the condition equations.

It is important to examine the relationships that link these various systems. Let $$- {E},$$ $$- {E}',$$ $- {E}'',$ etc., be such a system of corrections, different from the most plausible system. Then we have

Multiplying these equations by $${A},$$ $${B},$$ ${C},$ etc., and adding, we obtain, with the help of the equations (10),

But combining equations (13) in the same manner gives

From the combination of these results, one easily deduces

Therefore, the sum

is necessarily greater than the sum

which can be stated as follows:

The sum of squares of the corrections by which the observations can be reconciled with the condition equations, being multiplied by the weights of the corresponding observations respectively, gives a minimum, if the most plausible corrections are adopted.

This is the very principle of least squares, from which equations (12) and (10) can be easily and immediately derived. The minimum value of the sum, which we will now denote by $${S},$$ is equal, by equation (14), to

11.
The determination of the most plausible errors, being independent of $$l,$$ $$l',$$ $l'',$ etc., clearly provides the most convenient preparation, regardless of how the observations are to be subsequently used. Furthermore, it is easy to see that, to achieve this goal, it is not necessary to perform elimination indefinitely, i.e. to calculate $$(\alpha\alpha),$$ $(\alpha\beta),$ etc., it suffices to deduce from the equations (12), by a definite elimination, the auxiliary quantities $${A},$$ $${B},$$ ${C},$ etc., which we will call, in what follows, the correlates of the condition equations

and to substitute these quantities into equation (10).

This method leaves nothing to be desired when only the most plausible values of the quantities provided by the observation are required, but it is different when one also seeks the weights of each of the found values. Whichever of the four previous formulas one wants to use, it is essential to know $${L},$$ $${L'},$$ ${L''},$ etc., or equivalently, $$x^0,$$ $$y^0,$$ $z^0,$ etc.; for this reason, it will be useful to more closely study the elimination that provides these quantities, and to obtain a more convenient method for determining the weights.

12.
The relations between the quantities we are concerned with are notably simplified by considering the indefinite second-degree function

which we will denote by $${T}.$$ This function is obviously equal to

Moreover, we clearly have

and if, with the help of the equations (7), one expresses $$x,$$ $$y,$$ $z,$ etc. in terms of $$\xi,$$ $$\eta,$$ $\zeta,$ etc.,

The theory developed above provides two sets of determined values for the quantities $$x,$$ $$y,$$ $z,$ etc., $$\xi,$$ $$\eta,$$ $\zeta,$ etc.

The first one is

whose corresponding value of $$T$$ is

as can be seen by comparing the third form of the weight $${P}$$ with equation (16), or by directly considering the fourth form.

The second set of values is

and the corresponding value of $${T}$$ is

as is evident from formulas (10) and (15), and also from formulas (14) and (16).

13.
We must now subject the function $${T}$$ to a transformation similar to that which has been indicated in Theoria Motus Corporum Coelestium art. 182, and was further developed in Disquisitione de elementis ellipticis Palladis. To this end, let us set

After also setting

we will have

where $$\eta',$$ $$\zeta,$$ $\varphi',$ etc. are derived from $$\xi,$$ $$\eta,$$ $$\zeta,$$ $\varphi,$ etc. using the following equations:

We can then easily derive all the formulas required for our purpose. Namely, to determine the correlates $${A},$$ $${B},$$ ${C},$ etc., we set {{MathForm2|(18)|$$\left.\begin{array}{l}\begin{alignat}{6} &\mathfrak{B}'  &{}={} &\mathfrak{B} &{}-{}& \frac{(ab)}{(aa)} & \,\mathfrak{A} &, \\[0.75ex] &\mathfrak{C}'' &{}={} &\mathfrak{C} &{}-{}& \frac{(ac)}{(aa)} & \,\mathfrak{A} &{}-{} \frac{(bc,1)}{(bb,1)} &\,\mathfrak{B}' &, \\[0.75ex] &\mathfrak{D}' &{}={} &\mathfrak{D} &{}-{}& \frac{(ad)}{(aa)} & \,\mathfrak{A} &{}-{} \frac{(bd,1)}{(bb,1)} &\,\mathfrak{B}' &{}-{} \frac{(cd,2)}{(cc,2)} &\,\mathfrak{C}, \end{alignat} \\ \,\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdot \end{array} \right \}$$}} and then $${A},$$ $${B},$$ $${C},$$ ${D},$ etc. are obtained from the following formulas, starting with the last one: {{MathForm2|(19)|$$\left.\begin{array}{l}\begin{align} (aa)\,{A} + (ab)\,{B} + (ac)\,{C} + (ad)\,{D} + \ldots &= \mathfrak{A}, \\ (bb,1)\,{B} + (bc,1)\,{C} + (bd,1)\,{D} + \ldots &= \mathfrak{B}', \\ (cc,2)\,{C} + (cd,2)\,{D} + \ldots &= {C}'', \\ (dd,3)\,{D} + \ldots &= \mathfrak{D}''', \end{align} \\ \quad\qquad\qquad\qquad\qquad\qquad\cdots\cdots\cdots\cdots\cdots\cdots\cdot\cdot \end{array}\right\}$$}} For the sum $${S},$$ we have the new formula

and finally, the weight $${P}$$ which is to be assigned to the most plausible determination of the quantity $$u,$$ is given by the formula

In this formula, we have {{MathForm2|(22)|$$\left.\begin{alignat}{5} &(bl,1) &{}={} &(bl) &{}-{}& \frac{(ab)\,(al)}{(aa)}, \\ &(cl,2) &{}={} &(cl) &{}-{}& \frac{(ac)\,(al)}{(aa)} &{}-{}& \frac{(bc,1)\,(bl,1)}{(bb,1)}, \\ &(dl,3) &{}={} &(dl) &{}-{}& \frac{(ad)\,(al)}{(aa)} &{}-{}& \frac{(bd,1)\,(bl,1)}{(bb,1)} &{}-{}& \frac{(cd,2)\,(cl,2)}{(cc,2)}\cdot \end{alignat} \right\}$$}}

Formulas (17)....(21), whose simplicity leaves nothing to be desired, provide the complete solution to our problem.

14.
Having solved our primary problem, we will now address some secondary questions, which will shed more light on the subject.

We will first investigate whether it is possible for the elimination that provides $$x,$$ $$y,$$ $z,$ etc., as functions of $$\xi,$$ $$\eta,$$ $\zeta,$ etc., to become impossible in certain cases. This would obviously happen if the functions $$\xi,$$ $$\eta,$$ $\zeta,$ etc., were not independent of each other. Suppose, for a moment, that this is the case, and that one of them can be expressed as a function of the others, such that we have an identity

where $$\alpha,$$ $$\beta,$$ $\gamma,$ etc. are determined numbers. Then we will have

and if we set

it follows automatically that

and consequently,

where $$p,$$ $$p',$$ $p'',$ etc. are, by their nature, all positive, this equation requires

Let us now consider the complete differentials $${dX},$$ $${dY},$$ ${dZ},$ etc. corresponding to the values of $$v,$$ $$v',$$ $v'',$ etc., immediately provided by the observations. By our previous results, these differentials

will be related to each other in such a way that when they are multiplied by $$\alpha,$$ $$\beta,$$ $\gamma,$ etc. respectively and added, the sum is identically zero, so that among the equations

at least one can be regarded as superfluous, because it will be satisfied as soon as the others are.

Upon closer examination of the question, it is seen that this conclusion is only applicable to values of the variables that differ infinitesimally from those provided by observation. In particular, there are two cases to distinguish: first, the case where one of the equations

is generally and absolutely contained within the others, and can therefore be eliminated; and second, the case where, for the particular values of $$v,$$ $$v',$$ $v'',$ etc. provided by the observations, one of the functions $${X},$$ $${Y},$$ ${Z},$ etc., $${X}$$ for example, achieves its maximum or minimum value, or, more generally, a value where its differential vanishes, while the other equations remain satisfied. However, since we only consider variations of our variables whose squares are negligible, this second case (which in practice will rarely occur) can be assimilated into the first, and one of the constraint equations can be removed as redundant. If the remaining equations are independent in the sense we have indicated, it is certain, according to the above, that the elimination is possible. However, we reserve the right to return to this matter, which deserves to be examined as a theoretical subtlety rather than as a question of practical utility.

15.
In the previous commentary, arts. 37 sqq., we have shown how to approximate the weight of a determination a posteriori. If approximate values of $\pi$ quantities are provided by equally precise observations, and they are compared with the values resulting from the most plausible assumptions one can make about the $\rho$ elements on which they depend, it has been seen that one must add the squares of the obtained differences, divide the sum by $$\pi - \rho,$$ and then the resulting quotient can be regarded as an approximate value of the square of the average error inherent in this kind of observation. If the observations are of unequal precision, the only modification that must be made is that one must multiply the squares of the differences by the respective weights of the corresponding observations, and the mean error obtained in this way relates to observations whose weight is taken to be unity.

In the current case, the sum of the squares we are talking about obviously coincides with the sum ${S}$, and the difference $$\pi - \rho$$ with the number $\sigma$ of the condition equations. Consequently, for the average error of observations whose weight is $$1$$, we will have the expression $$\sqrt\tfrac{\sigma},$$ so that the determination will be increasingly reliable for larger values of $$\sigma.$$

However, it is worthwhile to establish this result independently of the reasoning in the first investigation. To this end, it will be helpful to introduce some new notation. Suppose that, corresponding to the values

we have

so that

and furthermore, corresponding to the values

we have

and finally, corresponding to the values

we have

and so on.

Combining equations (4) and (9) then yields

and, since

we will clearly have

16.
The series of observations that provide the quantities $$v,$$ $$v',$$ $v,$ etc., affected by random errors $$e,$$ $$e',$$ $e,$ etc., can be considered as a test that does not reveal the magnitude of each error, but instead, by means of the rules explained above, allows us to determine the quantity $${S}$$ as a function of the errors. In such a test, some errors may be larger and others smaller; but the greater the number of errors used, the greater the probability that $${S}$$ differs little from its average value. The difficulty thus boils down to finding the average value of $${S}.$$

By the principles outlined in the first Memoir, which it is unnecessary to reproduce here, we find for this average value

Let $$\mu$$ denote the average error corresponding to observations with weight 1, so that we have

Then the previous expression can be written as follows:

but we have found

and thus the right-hand side is unity, as can be easily recognized by comparing equations (6) and (7). Similarly, we find that

and so on.

Hence the average value of $${S}$$ is $$ =\sigma\mu^2,$$ and insofar as it is permissible to regard the random value of $${S}$$ as equal to the average value, we conclude that $$\mu = \sqrt\tfrac{\sigma}.$$

17.
One can assess the confidence warranted by this determination by calculating either the mean error to be feared, or its square. The latter will be the square root of the average value of the expression $$\left( \frac{\sigma} - \mu^2 \right)^2 ,$$ which can be developed by reasoning similar to that which was explained in the first commentary (arts. 39 sqq.). We will omit these for the sake of brevity, and simply give the result.

The mean error to be feared in determining the square $$\mu^2$$ is expressed by

where $$\nu^4$$ is the average value of the fourth powers of the errors with weight 1, and $${N}$$ is the sum

This sum cannot generally be simplified; however it can be shown, by a method similar to that used in art. 40 of the previous commentary, that its value is between $$\pi$$ and $$\frac{\sigma^2}{\pi}.$$ With the hypothesis on which we originally established the method of least squares, the term containing this sum disappears, because $$\nu^4 = 3\,\mu^4 ,$$ and the precision that must be attributed to the determination $$\mu = \sqrt\frac{\sigma}$$ is therefore the same as if one had operated on $$\sigma$$ observations affected by exactly known errors, following the precepts of arts. 15, 16 of the previous commentary.

18.
For the compensation of observations, we have seen above that there are two operations to perform: first, it is necessary to determine the correlates of the conditional equations, that is, $${A},$$ $${B},$$ $${C},$$ etc., which satisfy equations (12); secondly, we must substitute these quantities into equation (10). The compensation thus obtained can be called perfect or complete, as opposed to an imperfect or incomplete compensation. We will use the latter term to describe those resulting from the same equations (10), in which values of $${A},$$ $${B},$$ $${C}$$ are substituted which do not satisfy the equations (12), i.e. which satisfy only some or none. We will not consider such a system of corrections here, and we will not even describe them as a compensation. When equations (10) are satisfied, systems (12) and (13) become equivalent, and the difference of which we speak can then be stated as follows: Fully compensated observations satisfy the conditional equations

Partially compensated observations satisfy only some of these equations, and perhaps none; a compensation for which all equations are satisfied is necessarily complete.

19.
It follows from the very definition of a compensation that a combination of two compensations is again a compensation, and it does not matter whether the rules given to obtain a perfect compensation are applied to the raw observations or to imperfectly compensated observations.

Let $$- \theta,$$ $$- \theta',$$ $$- \theta'',$$ etc., be a system of incomplete compensations resulting from the formulas {{MathForm2|(I)|$$\left\{\begin{array}{l}\begin{alignat}{5} &\theta\,p  &{}={} {A}^0 &\, a   &{}+{} {B}^0 &\, b   &{}+{} {C}^0 &\, c   &{}+{} \ldots, \\ &\theta' p' &{}={} {A}^0 &\, a'  &{}+{} {B}^0 &\, b'  &{}+{} {C}^0 &\, c'  &{}+{} \ldots, \\ &\thetap &{}={} {A}^0 &\, a &{}+{} {B}^0 &\, b &{}+{} {C}^0 &\, c'' &{}+{} \ldots, \end{alignat} \\ \ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots \end{array}\right.$$}} Since the observations thus changed do not satisfy all of the conditional equations, let $$\mathfrak{A}^\ast,$$ $$\mathfrak{B}^\ast,$$ $$\mathfrak{C}^\ast,$$ etc., be the values taken by $${X},$$ $${Y},$$ $${Z},$$ etc., when the values obtained for $$v,$$ $$v',$$ $$v'',$$ etc., are substituted into them. We must then find the values $${A}^\ast,$$ $${B}^\ast,$$ $${C}^\ast,$$ etc., which satisfy the equations {{MathForm2|(II)|$$\left\{ \begin{array}{l}\begin{alignat}{5} \mathfrak{A}^\ast &{}={}& {A}^\ast &\,(aa) &{}+{} {B}^\ast &\,(ab) &{}+{} {C}^\ast &\,(ac) &{}+{} \ldots, \\ \mathfrak{B}^\ast &{}={}& {A}^\ast &\,(ab) &{}+{} {B}^\ast &\,(bb) &{}+{} {C}^\ast &\,(bc) &{}+{} \ldots, \\ \mathfrak{C}^\ast &{}={}& {A}^\ast &\,(ac) &{}+{} {B}^\ast &\,(bc) &{}+{} {C}^\ast &\,(cc) &{}+{} \ldots, \end{alignat} \\ \ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots \end{array}\right.$$}} Once this is done, the complete compensation of the observations thus modified will be accomplished by the new changes $$- \varkappa,$$ $$- \varkappa',$$ $$- \varkappa,$$ etc.; $$\varkappa,$$ $$\varkappa',$$ $$\varkappa,$$ etc., being deduced from the formulas {{MathForm2|(III)|$$\left\{ \begin{array}{l}\begin{alignat}{5} &\varkappa\,p  &{}={} {A}^\ast &\, a   &{}+{} {B}^\ast &\, b   &{}+{} {C}^\ast &\, c   &{}+{} \ldots, \\ &\varkappa' p' &{}={} {A}^\ast &\, a'  &{}+{} {B}^\ast &\, b'  &{}+{} {C}^\ast &\, c'  &{}+{} \ldots, \\ &\varkappap &{}={} {A}^\ast &\, a &{}+{} {B}^\ast &\, b &{}+{} {C}^\ast &\, c'' &{}+{} \ldots, \end{alignat} \\ \ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots \end{array}\right.$$}} Let us now examine how these corrections agree with the complete compensation of the raw observations. First of all, it is clear that we have

In these equations, if we replace $$\theta,$$ $$\theta',$$ $$\theta'',$$ etc. with their values from (I), and if we replace $$\mathcal{A}^\ast,$$ $$\mathcal{B}^\ast,$$ $$\mathcal{C}^\ast,$$ etc., with their values from (II), then we obtain

It follows that the correlates of the conditional equations (12) are

and then equations (10), (I), and (III) show that

Hence the perfect compensation produces the same value for each unknown, whether calculated directly or obtained through an incomplete compensation.

20.
When there are a large number of conditional equations, determining the correlates $${A},$$ $${B},$$ $${C},$$ etc. may require calculations so lengthy that the calculator may be deterred: in such cases, it may be advantageous to obtain a complete compensation using a series of approximations based on the theorem of the preceding article. For this purpose, the conditional equations will be divided into two or more groups, and first, a compensation will be sought that satisfies the equations of the first group. Then, the values modified by this initial calculation will be treated, and they will be corrected again, considering only the equations of the second group. This second calculation will generally yield results that no longer satisfy the equations of the first group, and if only two groups were formed, one must then return to the first group and satisfy its equations with new corrections. The observations will then undergo a fourth compensation, in which only the conditions of the second group are considered, and by alternating in this way between the two groups of equations, corrections will be formed that will necessarily become smaller and smaller. If the choice of groups has been made skillfully, one will arrive at stable values after a few iterations.

When forming more than two groups, one must proceed in the same manner, with the various groups being used successively until the last, after which one returns to the first group to repeat the process in the same order. We have only indicated this method, the success of which will depend greatly on the skill of the calculator.

21.
We have yet to provide the proof of the lemma which was assumed in art. 8. For the sake of perspicuity, let us use notation which is better adapted to this matter.

Let $$x^0,$$ $$x',$$ $$x'',$$ etc., be indeterminates, and suppose that the equations

have, through elimination, led to the following:

By substituting, in the first two equations of the second system, the values of $${X}^0,$$ $${X}',$$ $${X}'',$$ etc., provided by the first, we obtain two identical equations:

These equations being identical, we can substitute any quantities we wish for $$x^0,$$ $$x',$$ $$x'',$$ etc. Let's choose in the first

and in the second

Subtracting the two identities element-wise yields $$\begin{align}{N}^{10} - {N}^{01} {}={} &({N}^{00} \, {N}^{11} - {N}^{10} \, {N}^{01}) \, (n^{01} - n^{10}) \\ {}+{} &({N}^{00} \, {N}^{12} - {N}^{10} \, {N}^{02}) \, (n^{02} - n^{20}) \\ {}+{} &({N}^{00} \, {N}^{13} - {N}^{10} \, {N}^{03}) \, (n^{03} - n^{30}) \\ {}+{} &\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots \\[0.75ex] {}+{} &({N}^{01} \, {N}^{12} - {N}^{11} \, {N}^{02}) \, (n^{12} - n^{21}) \\ {}+{} &({N}^{01} \, {N}^{13} - {N}^{11} \, {N}^{03}) \, (n^{13} - n^{31}) \\ {}+{} &\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots \\[0.75ex] {}+{} &({N}^{02} \, {N}^{13} - {N}^{12} \, {N}^{03}) \, (n^{23} - n^{32}) \\ {}+{} &\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots \end{align}$$ This can be written more succinctly as: $${N}^{10} - {N}^{01} = \sum \left({N}^{0\alpha} \, {N}^{1\beta} - {N}^{1\alpha} \, {N}^{0\beta} \right) \left(n^{\alpha\beta} - n^{\beta\alpha}\right) ,$$ where $$\alpha,$$ $$\beta$$ denote two arbitrarily chosen indices; from this, we conclude that the equalities $$\begin{aligned}n^{01} &= n^{10}, & n^{02} &= n^{20} ,\end{aligned}$$ and more generally, $$n^{\alpha\beta} = n^{\beta\alpha}, \ldots ,$$ and so $${N}^{01} = {N}^{10} \,;$$ Moreover, since the order of the indeterminates is arbitrary, it is evident that, under the assumed hypothesis, we have $${N}^{\alpha\beta} = {N}^{\beta\alpha}$$ in general.

22.
Since the method outlined in this commentary is intended primarily for useful application in high geodesy, the reader may appreciate it if we include some examples drawn from this branch of science.

The conditional equations that exist among the angles of a system of triangles can generally be divided into three categories.

I. The sum of the horizontal angles formed around the same vertex, encompassing the entire horizon, must be equal to four right angles.

II. The sum of the angles of each triangle can always be considered as known; for even when the triangle is situated on a curved surface, the excess of the sum of its angles over two right angles can be calculated with such approximation that it is permissible to consider the result as absolutely exact.

III. Finally, a third kind of relation is obtained by examining the ratios of the sides in triangles that form a closed network. If, indeed, the triangles are placed in such a way that the second triangle has a side $$a$$ in common with the first, and a side $$b$$ in common with the third; if the fourth triangle has two sides $$c$$ and $$d$$ respectively in common with the third and the fifth, and so on, until the last triangle, which has a common side $$k$$ with the previous one, and with the first of all a common side $$l,$$ then the quotients

can be calculated by means of the angles opposite to them in the triangle whose two compared sides are part of it, and since the product of these fractions is obviously unity, there will be a relation between the sines of the various measured angles (reduced by one-third of the spherical or spheroidal excess when operating on a curved surface).

In somewhat complicated networks, it often happens that there are too many equations of the second and third category, and some of them are contained in the rest. On the other hand, it may happen in rare cases that some new equations must be added to those of the second category. For example, this will occur when the network contains polygons not divided into triangles by measurements; one must then introduce equations related to figures with more than three sides. On another occasion, we will return to this and provide more details on these various circumstances, but examining them now would divert us from our purpose. However, we cannot refrain from making an essential remark here for those who would like to make the rigorous application of our theory: we always assume that the quantities denoted by $$v,$$ $$v',$$ $v'',$ etc., have been observed directly, or deduced from observations, in such a way that their determinations are independent of each other, or at least can be regarded as such. In common practice, one observes the angles that can be considered as the elements $$v,$$ $$v',$$ $v'',$ etc. themselves. But one must not forget that if the system contains, in addition, triangles whose angles have not been directly observed and have been deduced from those that are known, by additions or subtractions, then these angles should not be counted among the quantities determined by the observation, and they must be included in the calculation as functions of the elements used to form them. It will be different if one adopts the method of observations of (Astronomische Nachrichten II, p. 431), which consists of determining all directions around the same vertex, by relating them all to a single arbitrary direction. The angles measured in this way will then be taken for $$v,$$ $$v',$$ $v'',$ etc., and the angles of the triangles will all appear as differences. The equations of the first category should, in this case, be suppressed as superfluous, for they will be identically satisfied. The method that I myself followed in the triangulations carried out during the last few years differs from the two previous methods; however, it can be reconciled, as to the result, with Struve's method, in the sense that, at each station, one must regard $$v,$$ $$v',$$ $v'',$ etc., as the angles formed by the directions emanating from it, with a single arbitrarily chosen line.

We will give two examples: the first relates to the first mode of operation, and the second is related to observations made according to the second method.

23.
The first example comes from the work of, Précis historique des opérations trigonométriques faites en Hollande. We will seek to compensate the part of the observations related to the system of triangles that are contained between Harlingen, Sneek, Oldeholtpade, Ballum, Leeuwarden, Dockum, Drachten, Oosterwolde, and Gröningen. Between these points, nine triangles numbered 121, 122, 123, 124, 125, 127, 128, 131, 132 were formed in the cited work. The observed angles are as follows:

The consideration of these triangles shows that the twenty-seven angles directly provided by observation have thirteen necessary relationships among them, namely: two of the first kind, nine of the second kind, and two of the third kind. However, it is not necessary to write down all these equations here in their final form, because for the calculation we only need the quantities which were denoted in the general theory by $$\mathcal{A}$$, $$a$$, $$a'$$, etc., $$\mathcal{B}$$, $$b$$, $$b'$$, etc. etc. Therefore, we can immediately write down the equations (13), which bring out these quantities. Instead of $$\varepsilon$$, $$\varepsilon'$$, etc., we will simply write here $$(0),$$ $$(1),$$ $$(2),$$ etc.

In this way, the two equations of the first kind correspond to the following:

Next, we find, for the spheroidal excesses of the nine triangles: $$1.749;$$ $$1.147;$$ $$1.243;$$ $$1.698;$$ $$0.873;$$ $$1.167;$$ $$1.104;$$ $$2.161;$$ $$1''.403.$$ Then, we have the conditional equation of the second kind:

and so on for the others, and we have the following nine equations:

The conditional equations of the third kind are more easily expressed using logarithms. The first one is:

It seems unnecessary to develop the other one in finite form. To these two equations correspond the following, in which the coefficients refer to the seventh decimal place of Brigg's logarithms:

No reason leads us to attribute unequal weights to the various observations, so we will assume

Denoting the correlates of the condition equations, in the same order as these equations were written, by

we determine them by the following equations:

We then deduce by elimination:

Finally, the most plausible errors are given by the formulas

and we obtain the following numerical values, to which we add, for comparison, the corrections adopted by :
 * }

The sum of the squares of our corrections is 97.885; the average error, as indicated by the 27 observed angles, is therefore

The sum of the squares of the corrections by is 341.42.

The triangles whose vertices, in the triangulation of Hanover, are placed at Falkenberg, Breithorn, Hauselberg, Wulfsode, and Wilsede, will provide us with a second example.

The following directions have been observed:

These observations allow us to form seven triangles.

We have here seven conditional equations of the second kind (there is obviously no need to form any of the first); to form them, we must first find the spheroidal excesses of the seven triangles, and for this it is essential to know the length of one side. The one that connects Wilsede to Wulfsode is 22877.94 meters. We conclude that the spheroidal excesses of the various triangles are:

If we denote the angles that determine the directions indicated above by $$v^{(0)},$$ $$v^{(1)},$$ $$v^{(2)},$$ $$v^{(3)},$$ etc., then the angles of the first triangle, marked with the same indices, will be

and the first condition equation is therefore

The remaining six triangles will yield six similar equations, but a little attention shows that these equations are not independent. Indeed, the second is identical to the sum of the first, fourth, and sixth; the sum of the third and fifth is identical to that of the fourth and seventh: for this reason we will ignore the second and fifth. Instead of writing the remaining equations in finite form, we will write here the corresponding equations (13), with symbols $$(0),$$ $$(1),$$ $$(2),$$ etc. taking the place of $$\varepsilon,$$ $$\varepsilon',$$ $$\varepsilon'',$$ etc.:

We can obtain eight conditional equations of the third kind from the triangles of the system, to which end one may combine three of the four triangles I, II, IV, VI, or three of the triangles III, IV, V, VII; however, a little attention shows that it suffices to consider two belonging to each of these two systems of triangles, and those will encompass all the others. Thus, for the sixth conditional equation, we have

and for the seventh,

to which correspond the equations (13),

If we attribute the same certainty to the various directions, setting

and if we denote the correlates of the seven conditional equations by $$ {A},$$ $$ {B}, $$ $$ {C}, $$ $$ {D}, $$ $$ {E}, $$ $$ {F}, $$ $$ {G},$$ then their determination will depend on the following equations:

From this, by elimination, we deduce

and the most probable errors are given by the formulas

from which we derive the following numerical values:

The sum of the squares of these errors is equal to $$1{,}2288;$$ hence, the average error resulting from the 18 observed directions is

25.
To give an example of the last part of our theory, let us determine the precision with which the compensated observations determine the Falkenberg-Breithorn side, using the Wilsede-Wulfsode side. The function $$u,$$ by which it is expressed, in this case, is:

Its value, deduced from the corrected observations, is

Differentiation of this equation provides the following, with $${d}v^{(0)},$$ ${d}v^{(1)},$ etc. expressed in seconds,

From this we find that:

Taking the meter as the unit of length, the methods indicated above yield,

We conclude that the mean error to be feared in the value of the Falkenberg-Breithorn side is $0{,}2886 m$ (where $$m$$ denotes the average error to be feared in the observed directions, this error being expressed in seconds), and consequently, if we adopt the value of $$m$$ announced above, this mean error to be feared is $$0^{\text{m}}1209.$$

Moreover, inspecting the system of triangles immediately shows that one cans completely disregard the Hauselberg station without breaking the network that connects the other four. However, it would not be good practice to eliminate the operations related to this point, as they certainly contribute to increasing the precision of the whole. To show more clearly what increase in precision results from this, we will finish by again performing the calculation, after excluding all results related to the Hauselberg point. Of the eighteen directions mentioned above, eight then cease to be used, and the most plausible errors on those that remain are: The value of the Falkenberg-Breithorn side then becomes $$26766^{\text{m}}{,}63,$$ a result little different from that obtained earlier. But calculating the weight yields

and the mean error to be feared, in meters, is

We see that by adding the operations related to Hauselberg, the weight of the determination of the Falkenberg-Breithorn side is increased in the ratio of $$7{,}644$$ to $$12{,}006,$$ that is, in the ratio of unity to $$1{,}571.$$