220 Mr. R. Meldrum Stewart on the 



quantities, and therefore Nn observation equations * ; the 

 equations (1) are condition equations f, expressing con- 

 ditions which are exactly fulfilled by the true values of 

 all the quantities involved, and must therefore also be 

 exactly fulfilled by the most probable values to be deduced. 

 If we denote the actual results of the observations on the 

 quantities w u y Xi z l9 etc. by #/, y/, z/, etc., the Nn obser- 

 vation equations will be 



QC] — X\ 



V\ = V\ 



*a — *i 



X% — «£ 2 



y* = I/*' 



^2 — -2 



. 



• 



. 



x^= Xn' 



sfv-y-s' 



2- N =2' N / 



stc. 1 



(2) 



J 



There are then in all (including the observed quantities 

 and the unknown constants a, b, c . . .) Nn-\- q unknown 

 quantities whose most probable values are to be deduced 

 from the Nw observation equations (2) in conjunction with 

 the N condition equations (1). 



Since, however, the equations (1) are non-linear, they 

 must first be reduced to the linear form. To effect this, we 

 deduce by some approximate method quantities A, B, C . . . 

 X 1? Y^ L x . . . X 2 , Y 2 , Z 2 . . . etc., which are reasonable 

 approximations to a, b, c . . . asi, y i9 z^ ... x s , y 2 , z 2 . . . etc., 

 and put 



a = A+A« tT 1 = X 1 + A* , i ,yi = Y 1 + Ayi etc. 

 b = B -f A/> x 2 = X 2 + A# 2 v 2 = Y 2 + Ay 2 



where Aa, Afr, Ac . . . A,! 1 !, A%-2, Aa' 3 ... el c. are corrections 

 to the approximate values A, B, . . . X 1? X 2 , X 3 . . . etc., 

 and are so small that for a first approximation their squares 



* An observation equation is a statement of equality between an 

 observed value and an expression which in its simplest form may 

 be merely the true value of the observed quantity, or on the other 

 hand may be a function of one or more quantities whose values are 

 required. Thus to every observation there corresponds one observation 

 equation, and vice versa ; and further, an observed value can enter an 

 observation equation only as a linear quantity with coefficient unity. 

 It seems desirable to stress these points, since they are sometimes 

 o\erlooked ; in fact it appears to be a certain confusion on this subject 

 which has led Dr. Campbell astray in his solution. 



t For a very sensible remark on the distinction which should be 

 observed between the terms "observation equation" and "condition 

 equation" (terms which are often confused), see Merriman's 'Theory 

 of Least Squares ' (original edition), footnote to Art. 56, p. 60. 



