Now, differentiating equations (4), we have: == a., dv, ay =-- bi, Ovi " = Ova 6v, etc., etc., etc.
so that (7) become: auri alvs + (sive + etc.= 0, (8) bit% + bit, + ba% + etc.-- 0, + Cu, civ, + etc.= 0.
It will be noticed that the number of equa tions of the form (8) is the same as the number of unknown quantities x, y, z, no matter how large may have been the number of original equations of the form (4). We shall therefore simply substitute in equations (8) values of the v's from equations (4), remembering that, ac cording to our notation: [an] = ma, + a,a. + etc., fab)=-- attri + a,b, + etc., [se] = arcs + ries + etc., etc., etc.
and thus obtain the following equations: [aalx [ably + [ads + [an] = 0, (9) [ob]s + [bcjs [1m) = 0, [ads [bc)y [c]s [en) = 0.
Equations (9) are called °Normal Equa tions.° As we have just seen, their number is always equal to the number of unknown quan tities, and their solution by elimination offers no ambiguity. Furthermore, owing to the method by which they have been deduced, it is clear that they will furnish for the unknowns x, y, z a system of numerical values, which, when substituted in the original equations, (4), will make the sum of the squares of the re sidual errors less than would be the case with any other possible system of numerical values for the unknowns, no matter how much other system might have been obtained.
If we now compare the above normal equa tions (9) with the original equations (4), which are usually called equations,D we see at once the law of formation of (9) from (4). It is evident that to form the normal equations we may use the following simple rule, first writing zeros instead of v's for the right hand members of equations (4).
Rule for Normal To form the first normal equation, multiply each obser vation equation throughout by the coefficient of its .v, and then add the resulting equations. To form the next normal, use in the same way the coefficients of y; and for the third nor mal, use the coefficients of a.
The normal equations once formed, they can be solved by any method of algebraic elimi nation; but it is generally most convenient to use the following special form of computation, due to Gauss.
From the first equation (9) we have: lab] [ac) [an) x[aa] 7— foal — [oar If we substitute this value of x in the other two normal equations (9) we shall eliminate x.
This substitution can be systematized by the introduction of auxiliary quantities [bb. 1], [bc. 1], etc., defined by the following equations: [ab [bb. 11 = [bb] — [ab] (bc. 11 =1bc1 — (-- , cab], ,,1[bn. 1) = [bnj — — an lac] [cc. 1] = [cc] — [at] [ca. 1] [cn) --(an].
[ad] Using these auxiliaries, the substitution of the above value of x gives: [bb. 11y [bc. 1]z [bn. 1] = 0, (11) [be. l]y + [cc. lis [cn. 1] = 0.
These equations, while precisely similar in form to the original normal equations (9), now involve one less unknown quantity, and their number has been reduced from three to two. A second application of the same method of elimination gives the following asecon& auxil iaries: [cc. 2] [cc. 1] — [be. 1].
(l2) [cn. 2) = [cn• I [bc. 1] [bb. and the equation: (13) [cc. 2]z +Eat 21=0.
From this equation. (13), we have for the value of z: (14)z [cn. 2] [cc. 2]' We now find the value of y by substituting this value of s in either of the equations (11) ; and x by substituting the values of y and a in any one of the normal equations (9).
Case of a Single It is of spe cial interest to consider the application of the Method of Least Squares when there is only a single unknown quantity, directly observed. An example of this, for instance, would be our measurement of a room, where the only un known is the width of the room, and several independent direct observations have been made.
Denoting the single unknown quantity by x, the observation equations of the form (4) will be: X + ft = 01, (15) x + n2 = V2, X Vs, etc.; and if we suppose there are in all m such ob servation equations, the (rule for normal equa tions( gives a single normal equation of the form: (16) mx 4-[n]'=0, and this will be the only normal equation. The solution is: (17) x or, in other words, the most probable value of x is the arithmetical mean of the several ob served values represented by the n's. This is a further strong addition to the evidence of plau sibility attaching to Legendre's theorem of Least Squares. Possibly no other simple admissible theorem as to the residuals would give this re sult for a single unknown, observed directly; yet this result is the only one consistent with common sense.