THIED ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 287 



To connect up with the order /A' = 4n -f 2 = /j., denoting p (/) by p', 



P' + I _ A K) - s (6v) _ (1 - 2m + 2m 2 ) ft - (I - 3m + 3m 2 ) 

 p' - 1 v (,,) - *(.V) (! - m)(2/3 - 1) 



_ 

 a 4 - 1 + 2,-< 6 4 



/= - e .- 



(e - 1) />- L 6* . 2a 3 + n* - ( I + 



We have found 



dn/K' = t ._-6_(l-/)K' 



o a + b H (fK.') 



and proceeding with the series, writing f for fK', 



dn3/_l-m _l-l> 



m 



b 



Working these out with the assistance of the analysis given subsequently, 



dn5/_ 3 - (1 -f a <r) IS + nk (a - n- 3 + l> ] ) , . 



o ~ a? (1 + a a-} />' nl> (H a- <? -f //) 



and generally 



dn(-2-+ 1)/_.E,,. + , +6F, r+1 a , 



" E, +1 - fiF^, 

 where F is a rational function of a and //', and E derivable from it by the substitution 



0,1 

 Thus for 



^ = 8^ + 4, ^(2^+l)/ =1; FB+I = O (45) . 



The results in the sequel give 



F 7 = a 4 (1 + a - 2a 2 - 3 ) - (a + 2 - a 3 - :5" 4 ) // - a 2 l>* (46), 



and therefore 



E 7 = a 5 - (3 3 + 4 - a 5 - a 8 ) b* + (1 + 2a - a 2 - 3 ) 6 s (47), 



F 9 = a (3 + - 3a 2 - a 3 ) - 3 3 (1 + 2a - 2a 2 - 2a 3 ) 6* 



+ - a 6 ) // + a 3 ^ 12 (48), 



