a ,  x&<_—_——— 
ON THE FURTHER TABULATION OF BESSEL, ETO,, FUNCTIONS. 43 
Or with @ between s, and 83, $9>8, 4>83, when a — 8 can vanish, 
and the integral becomes infinite 
1/3 ds r A(q + 7) 
a SS ae 1 as LUE 
(31) (ES See ~ op KB + bles 5A, 
(3 Jz ds r B(r — q) 
2 Ca Gee 1 tien! 
(32) [ —o« VS (1 50) KE(q) + 3108 B(r 4+ @) 
where F 
GpTSha 2) gn 8 Sarai A ea. 9 Bicisn F) ae a 
(33) ay sn °K, ae en?/K, aie: aay Rg = SUT. 
Putting AK =u, fK =v, these formulas are obtained by a double 
integration with respect to w of the relation 
— 4’ snvenvdny snucnudny , 
(1 — x? sn? sn?w)? , 
(34) dn? (v + uw) — dn? (v—u) = 
the first integration gives 
envdnv 
(35) Kam (v + u) + Ham (v — wu) = C— Tore 
(36) Eam (v + uv) + Ham (v— uw) — 2Eamv 
envdnv 
—gengdne _ sn v ; 
La sn v T—sn’vsnu ’ 
or, in Jacobi’s notation, f 
— 2«?snvenvdnv sn2u 
1 —x«*sn?v sn?u, 
(37) an(v + u) + an(v—u)—Aavs= 
Defining Jacobi’s Theta function by 
dlog@u _ Ou _ Jan udu 
(38) a a th e674 
and integrating (37) again with respect to ~ 
(39) log @ (v + w) — log @(v — u) — 2uznv 
| gd eased 
T 1 — «2 sn’v sn? 
du, 
0 
and this integral is Jacobi’s standard form of the Third Elliptic Integral, 
and denoted by II. (w, v). 
Then, with the notation 
envdnv_ d 
40 Z3v= pee tte I | 
(40) 23v=2nv + ae dy °8 H(v) 
( envdn v 
1 cee BE => i. 1 @(v — ) 
(4 " * 1 — x? sn?v sn727 Cabs 96:28 ¥ rh blog @(v + u)' 
