64 On certain Definite Integrals, [Jan. 8, 



This may be written 



K4)x^) tt+ K4r^S tt+7 (4rH) a+ • ■ • 



Hence if Ax m y n be a specimen term, we have, substituting for u, 

 (a^ + /3^- 1 + 7^- 2 + . . . )m=an v + bn v - l + cn v - 2 + . . ., 

 whence we have 



ari" + bn v -^ +cn"- 2 4- . . . 



m—- 



xni x + (3ni J -- 1 + ^n L - 2 + . . . 

 Hence we have to reduce 



an v +bn v ~ l + cn v - 2 -+ . . . 

 am* + $nv-- l + cn*- 2 + . . . 



to the form /VQ n d9. This is easily done, for the function is equivalent 

 to 



log 6 x logr^ \ g ex 

 6 ri+«in + r a +«2» + r 3 +«s» + 



Now we have 



c r \ t£i!Lr »- lr ^— 



f r t +'m — 2 ^/ ~ 1 € 4log e * CW, 



wlog e aJ-< 



2 Poo 



and ^ w ___( r +g n ) "e-^+^'^'dv. 



V 7T Jo 



Hence we see that the function can be reduced to the form 



(A + B?2, + Cw 2 + . . . + En*)ffFQ n ded<p.ffP l Q l n d0 1 d<j) 1 . , 

 Now if F»=M» a + Na^+PajYH- . . . 



(x—\ r Fx=M* r x« + N8 r xP+ . . . 

 \ dxj 



and consequently we have, if we put 



M=A/J . . . PPi . . . F(QQ ] . . .)<W0 • • • 

 + B/7 . . . PP X . . . F'iCQQi . . .)^# • • - 



