THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 249 



and now the change of m into 1 m and a into a is equivalent to a change of 

 sign of />, leaving ft unchanged. 

 Thus we find, with 2ft I = e, 



N n = 



N M - D n = 2>{/>* - e(e* + !);> + ^ - e + I)}, 



N n + D M = 2e|(e- 2)(e 2 - e- I)jp++ e ( c 2_ a) p 2 + e ** it 

 = 2e](e 2)y7 2 -f- ej -}(e 3 e l)p a + ef. 



Thus, if fji =. 22, .s(l I* 1 ) = s(a>), and N n = 0, which replaces in a much simpler 

 form the relation y 32 = 0. 



So also DH = makes ,s-(llr) = 00,7- = "" , and so enables us to connect up the 



results for /JL = 11 ; and other values of [L can he treated in the same way. 



23. There are three cases to consider, relative to the half-period 01, to which ^v is 

 congruent : 



T / 1 \ / \ i 3 72 /i\ 



and introducing WEBER'S function fat, or KTEPERT'S equivalent function L(2), 



K 4 1 6 (,s-, s. )' 2 1 8(1 2m) a 



(^/ == (yi w )~ = ^ /e ~ = 7~ ~w \ == ^7 w i 1 \ \^/' 



and to the complementary modulus :', 



_1_ 1 1 + 4a y/{ 18(1 -2m) a ,gv 



cn 2 /K" dir/'K' = 2 



cn2/K' = ]) , dn2/K' = -, sn (1 - 2/) K' = 6 (4) ; 



c c ot 



II. O) = at,, 



16 16i,-*,) 8 _i + 8(l-2m> = _. L(2)M 



W ~ (s, - S Z ) (*, - *,) " a 2 ( - m) (a - n + 1) 



, , (8); 



VOL. CCIII. A, 2 K 



