THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 251 



24. When p. is of the form 4n + 2, so that, as required in dynamical problems, 



then ^ p.v = (2n -f- 1) v is congruent to co, or <a.,, according as r is even or odd ; and 

 the integral I (r) is expressible in the form 



I (cr s) + x j& 



<T .S ^/S 



e -1- P -' -I- -1- P 



1 " //(- .v) 



and when the quantities a, wi and therefore a, b, c, cr, x, and P (r) have been 

 determined as algebraical functions of a parameter, the calculation of P,, . . . P M , 

 Q,, . . . , Q,,, is effected readily by a differentiation and verification. 

 It will be found that 



Q, = (2+ 1)P() (3), 



also 



P (w, + /O).,) / f.^ 7 , /\ P ( /Wo) ,-Tr/ 



// \ = zn (./ K ' * ' ~// \ = zf i/ K ( 4 )> 



v/( A 'i - *s) \/( s i ~ *s) 



znw denoting JACOBI'S Zeta function Zu or log <s)?t, while 



du 



ZMI = znit + log sim = log llx. (5) 



in GLAISHEH'S notation ; also 



cr x (w) "a -in 



Here already the degree of the results obtained by ABEL'S method have been 

 halved in ( 1 2) ; and the degree can be halved again by the substitution 



o cr x /-\ 



and now 



1 .1 s" + P,*"- 1 + ... 



- cos ' ' - .- 



2n + 1 (cr s) y 



- cos 



+ 1 r 



2_- sin-i rlr^/"^^^ (8). 



2n " 



2 K 2 



