ON THE FURTHER TABULATION OF BESSEL, ETC., FUNCTIONS. 41 
Jacobi has shown that the Zeta function is connected with the Theta 
function by the relation 
_- dlog @(hK) derbi gta 
(14) a hK = ae ee KEK(r) a log D(r). 
In a physical problem, as of Hlectro-magnetism or Dynamics, it is 
the Elliptic Integral which makes an appearance; and the physical 
student will be anxious to arrive at a numerical result without delay 
by utilising a Table of the Function. 
The integral which arises can be made to depend on three standard 
forms, called the First, Second, and Third Elliptic Integrals (I., II., and 
II. E. I.). 
These three integrals can be reduced to depend on the differential 
elements 
ds ds 1 ds 
15 Bitil — i 
(9) Wie act Ch @Bhice se V8 
where § is a cubic in the variable s, which may be written 
(16) S = 4 (s — 8)) (s — 82) (s — $3) 
when resolved into factors. 
Normalised to a standard form, of zero dimensions, the three 
integrals are written 
8 
J (s,; — 83)ds { s—oa ds L/=a ds 
Se eo TL) |2-— = 
a) (OS, any (52% ay [Sy 
$3 
3 denoting the value of S for s = a, and assuming a sequence 
(17) co > S> 8) > 8S: > 8 > 53> — 9. 
Taking, for example, the interval s.>s>53, the substitution 
(18) S$ — 83 = (Sq — $3) sin’¢, 
S_ — $ = (8 — 83) Cos’9, 
sy == s= (s — $3) Ad, 
(19) sin? $3 “$3 Gott — ae $2 
5); — 83 8, — S83 
reduces (I.) to Legendre’s standard form in (1): 
: $ 
/ (8, — 83)ds _ (do 
20 Rete a) eer : 
(20) | Js - Fg, or hK 
where = = 
J (s,;—83)ds d 
21 ee eae a ee 
(21) | = | : 
0 
Then (II.) becomes 
s 
$ $ 
99 (81 =) — (8; —s) ds ee. | etd dp 
(22) | J (8, — 53) J5 fi a sa ieee 
$3 
= =" FQ) — BE), 
$, — 83 
