42 REPORTS ON THE STATE OF SCIENOE.—1912. 
and so depends on Legendre’s Second Elliptic Integral E(@), as well 
as F(9). 
Legendre has shown that (III.), taken as a definite ney ake between 
the limits s, and s;, can be expressed by F(¢) and E(#), K and EH, by 
means of the equation (p') of his ‘ Fonctions elliptiques, I., p. 141; 
thus, taking the sequence 
(23) Co > OPS, P89 >8 >83 > — oO, 
so as to avoid an infinity in the integral, 
i J= ds 
24 
( ) [as Fa ee 
and similarly, by an seach ati of s and oa, 
2 2V8 do = —— p) — 
(25) jis &° = K m iK = K Bly) — BF(), 
but now 
(26) Cais bit 1 a dies Sere A*y Oo ate cos*y 
S,—8, «?sin?y’ s,—s,  «*sin?y’ s,—s,  sin’y 
B(x) =fK, 
but @ and hK are given as before in (18), (19), (20). 
The function zs fK in (24) is defined by 
(27) KasfK =KanfK + [ee ee 
= KE(x) — EF(x) + tinge 
= KE(x) — BE(:) + Ky /( ase — ss) 
(« — $3)(8, = 53) 
But for a corrected Integral of the III. kind, between the limits 
s, and s, the Jacobian Theta function is required ; and 
(jos tan Loe UF = WK 
(28) \; VEO, = Kaa WK + blog 5a 
while 
“That Bonde ives 11, Of — WK. 
(29) = Og = WK an fK + blog Ses ia 
=2 KE) + } log PL) 
in the notation of the Tables, with 
(30) Fi 908) GO. 
