﻿IN THE METHOD OF LEAST SQUARES. 189 



For tlie final equation for a\ 



« e, + «^i' + = 0. 



For the corresponding final equations for Xq, which must be rigorous, we make either 



a io fo + a' w' «'o + = a jp Co + a' '"' e'o -|- 



or 



a fo -{-a' e'o + = aeo + a' e'o + 



according as the first (I.) or the second (II.) form of combination is adopted. 



e^, e, and e, are the values of e when the indeterminates Xq, j/o > *'> 2/ 



x\, ^1 , &c., replace x, y in (7). 



The final equations for the combuiation (I.) are : — 



F x-\-P'y + P"z-\- + L = 0, 



P'x+Qtj-^Q'z + + .l/=0, 



^ '' P'x+Q'y+Q"z-\- + N = 0, 



P = ic a a -}- iif a' a' -{- , Q = tc h h -\- w' h' b' -\- , R = 7e c c -\- w' c' c' -{- 



(9.) P' z=2cab-\-iD'a'b' -{- , Q' = le b c -\- w' b' c' -\- , R = w c d -^ w' c' d' + 



From the conditions I. and II. applied to the original equations (7), if we make 



^«=aic4-Sa, B ^ = b u- -{-^^ , Cy=cw-\-8y, 



Au ^a'w' ^8a' , B ^ = b' id' -\- 8 ^' , Cy' = e' io' + 8 y , 



may be obtained 



P (x-r,) + ^' (y-3/i) + + L + Px. + P' ,,. + = 



L + Pi-, 4-P'!/'i + = — eiSa — e'.aa' — 



Hence, 



P (x — .r,) + P' (y — y.) + =. 5« e, + 8«' e', + 



(10.) -P' (-C - .^i) + Q (y — y.) + = S ,3 ei + 8 (3' e'. + 



And in a similar manner, 



P (.To — x) + P' (yo — y) -}- = a w e„ -\- a' w' e'o -\- 



(11.) P' (xo — x) + Q (yo — y) + = b w eo -{- b' V}' e'o + 



Since e is the probable value of iVo ^~ '^N ^^id ij" the probable value of (x — a\), to ob- 



VOL. TI. NEW SERIES. 25 



