338 REPOKT— 1873. 



.r, - rt, ■ (.r, - a; J(.v, - xj ^. - a, ■ (-^i - ^3)(^, " ^a) J ' 

 Hence also 



(^l-«6)(-^'l-«l)L'x^l K-«6)(^2-«l)L'^2 ('■«^3-«6)(^3-«l)L'^3j 



It only remains to determine the value of v'iC'^i — ''^'l)('''^~*'2)(^l~•'^'s)}■• 

 ^or this purpose, let u\, u\, u\ be what u^, u.^, u^ become when we sub- 

 stitute x\, x\, .r'g for x^, x^, x^. 



Hence we have from Abel's theorem, applied to equation (3), (e„= +!)> 



r^'' Vx dx C'''^ P.r dx C'''' P(^) dx 



J''" P.r dx C"' Fx dx r "' Fx dx 



x-a'2\^B^^'^^'J „ x-a'2\/^''^\\ ^ x-a'2\/m' 



J^' P.r (?.r r""' P.r _^^ r"^^ Pa? f7:r 



6 



=0. From this it follows that n', = ?f J + Kj; and therefore 



6 



cd{i(,\....\ = (d(ii^ + K^, ....\ (6) 



Now al(u\ . . . . )i differs only by a constant factor from 



hence, comparing (5) and (6), we perceive the truth of (4), where a = l and 



/3 = 6. It is easy from this to see that (2) must be generally true, when 



^ . ^ . . 



we give K, the positive sign. If we give Kj the negative sign, we must 



change the signs of e^, e^, e^, &c. in the equation derived from Abel's theorem ; 

 but this may also be effected by changing the sign of \^}i{x) in the second 

 line of that equation, and therefore the signs of V^Rx^, V^Rx^, 'V^B-r^ in (5) ; 



hence al (m^ + Kj . . .), al(u^ — Kj) are equal, but have opposite signs. But 

 now put in equation (3) 



x'\ = a^,x\=a^,x'\ = a^, 

 then 



(x - rt,)(.^• - a^){x - a.Xx - « J(.r - a J(a7 - a J - {x - aj{c^ + c^x+c^)'' 



= G^- - ■^•'i)('^^ - 0('^ - OG"^' - •''■i)G^- - ■^JG'*^ - '■^3) : 



hence, putting x=(i^, we see that 



^(«0 - '^'\X«0 - ■'^'2)(«0 - '^-'3) ^(«0 - ■^'iX'^O - '^-JC^O - ■•^'3) 



