THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 221 



and _(p2i'-pr) 8 ,,,, 



>* 



iv'r 



_ 

 x- c/n " V2r Vi' " iy'r 



= I 8// (6), 



so that 



N IO = 2** 



-!- - j.t'O 8y) /" + (4,r 3i/ 3?/-) .' 

 y I 005 d. ; j 



Thus, putting ), = 2, 3, 4, . . . , 



2P(r)-P(2r) =J(l+y) (8), 



3P (i>) - P (3r) =1 (9), 



4P(r)-P(4r) =1(1 +//) + ' (10), 



5P()-P(5r) =l+ v ^ c <'l), 



6P () - P (Gr) = 1 (! + ;'/)+ I + ? ^ ( ~_; < 'Jo ( ' ")' 



7 P( / ,)_P( 7/ .) =2 + ? / + ?/( ' / " (13), 



/ 7 



8P() - P(8t.) =i(i + y) + 2 + ^ x) " ( ^ ''' ~ r) ( 14 )' 



9F(r)-P(9r) = 3 + '*' ( ' y " C 5 )> 



7;i 



fO + Nlu ( lf) )' 



Tiu 



+ .r(-2 ? /' + 4 / /'+r) -/'-?/, 



(17), 



- 2 



