224 ; Mr* R. Meldrum Stewart on the 



He would thus have been confronted with exactly the same 

 difficulty as experienced by Dr. Campbell in the problem of 

 the determination of density *. The first solution corre- 

 sponds to the suppositious case in which the abscissae have 

 been measured with absolute precision, while the measures 

 of the ordinates are subject to accidental error ; the second 

 to the converse case. When both the co-ordinates are 

 subject to errors of measurement neither solution is correct. 

 Denoting, for the moment, the numerical measured values 

 of the co-ordinates by Xy. x 2 . . - y\ , y 2 • • ■ we have the 

 eight observation equations 



Xl — x-l' y x = *//, 

 x 2 = x 2 ' y 2 = y 2 ', 



#3 = «»' vz = yi> 



x± = x 4 y i = y± , 

 and the four condition equations 



y t = S^ + T, 



y 2 = Sa? 2 + T, 

 y 3 = Sa,+T, 

 y± = Sa? 4 + T, 



from which to determine the most probable values of ten 

 quantities, the eight observed co-ordinates and the two 

 constants S and T. 



To reduce the condition equations to the linear form 

 we put 



S = S -^-AS, 



x 2 = X 2 + Ax 2 , 

 x z = X 3 + Ax 3 , 



x 4 = X 4 + A# 4 , 



where S , X l5 X 2 . . . are approximate values and A!S, Aa? l5 

 Ax 2 ... the most probable corrections to these. Neglecting 

 products of the latter, the condition equations become 



yi = S^i + XiAS + T, 



y 2 = S a? 2 + X 2 AS + T, 

 etc. ; 

 and substituting in the observation equations these take 

 * Phil. Mag. he. cit. pp. 192, 193. 



