Theory of the Method of False Position. 661 



Multiplying and summing wo have 



the symbol S' being interpreted as before. 

 Lastly, solving equations (vi.) we have 



If results of which (viii.)-(x.) are the types be substituted in 

 iv. we have n equations to find the unknowns A a, A /3,. . .A v. 



These equations did not look very hopeful oh initio. I solved 

 them, however, by brute force for the first three cases, or for 

 formulae involving only one, two, and three constants, and to 

 my surprise the results came out with remarkable simplicity 

 of form — namely, the general regression equations discussed 

 in my memoir of 1901 (Phil. Trans. A. vol. 200. p. 9). A 

 little consideration showed that the analytical process was 

 similar to that involved in the discussion of the theory of 

 multiple correlation, but there seemed to be no direct physical 

 reason for applying the results of the correlation theory to 

 the problem of false position. 1 therefore put equations (iv.) 

 and (viii.)-(x.) before Dr. L. N. Gr. Filon, who has so often 

 come to my aid in algebraical difficulties, and he has provided 

 me with the following general solution. 



We have, using X aa to denote S(c« 2 ), Xj3j3 fo1 ' S J </)> 

 Xa(3 for S(cac^), &a, and f 0a for S(c«0/ o -?/)), ^ u/3 for 

 ^( 6 '/3(3/o — j/))? &c. from (ix.) and (viii.) : 



m ( p= n ( \ 



X>ea = pT 2 J ^ {dl^dpaO-p 2 ) +$ / ((dpedp'a + dp'edpa)o r 2)0~p'>'pp') \ 

 ill ( P- n -\ 



^ 6/3 ~ D 2 1 ^ {dpedppo-p 2 ) +8 / ((dp e dp'p t dp'edp/3)o-po-p'rpp f ) i 

 in ( p =n 



%ee = ™ \ ^ (dpe 2 G2f) +2$'((dpedple)o-p<Tpirp / /) \ 



* €V ~ W 2 { ? (<hedpv<rp) + 8'((dpt:dp' v + dp'e.dp V )(rp(r p > rpp') 1 



Multiply by A s a, A s /3...A s v respectively and add. remem- 

 bering that 



S {d pe A S €)=Qj or =D, according as p is not or is equal to s : 



e — a 



<•:-)- mosP = » 



& (X €e 'A S €') = jy S (dp&prsp). . . (xi.) 



Phil. Mag. S. 6. Vol. 5. No. 30. Jime 1903. 2 Y 



