THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 275 



and 



P(w) = a; 1 zn/K / (10), 



s - x a ~ 



n (8 - X 3 )*" 



= l s}n - 1 Iio-IV- + . - - +(- tJ-'R.-,*-' /r - 

 n (8-,T-)i vt^i 



Thus for the vector of the corresponding herpolhode 



M p = ^/(S - a; 3 ), x = x.sn (K - nit) (12), 



M **-->' = {(R + ll lX + ...) v/^X, + i (R,, - H,x- + ...) v/iXj- (13), 

 and in the associated hodograph described by the axis of a toj) 



sn 



= {(A +A 1 x+...A 2/( _ 1 ^-'),/J-X 1 +i(A -A r T+...-A ;! ,,_ 1 ^'- | ) v /iX,}' (14). 



36. With /u, = 4n, the resolution of r f into factors for the spherical catenary must 

 be such that Tj and T 2 are quadratics in t~ ; and thus 



T= 



(15), 



Tl = _ (k _ 1)* + 2 2 ~ - F + i ^ 3 - (k + l) 2 ^ 1 (17), 



T , = (k + IY-2 (2 ^l - A* + l) $ + ^ - I) 2 ? (18). 







h 



2 N 2 



