44 REPORTS ON THE STATE OF SCiENCE.—1913. 
and this integral is converted into the form (III.) by putting 
(42) sn?4 = 283, en’y = 8278, dnt = S17 8, 
Sy — 83 So — S83 8; — 83 
(43) ee we Road ue BaF ag ony eee 
ksn?y  8.—583 «sn 82. — 583 sn*y 8, — 83 
with the sequence 
(44) co > a> §) >So > S> 83 > — co, 
So also for the other forms, and thus all these integrals can be made 
to depend on the numerical value of a function in these Tables, with 
the addition of an algebraical or logarithmic function in the general 
case, 
But, unfortunately, in the physical problem it is the circular form, 
so called by Legendre, of the Third Elliptic Integral which is required ; 
here « lies in the interval s; >o> 5s), or 8; >o> — co; so that = is 
negative, and the standard form must be changed to 
4 aV(— 2%) ds 
(45) | A=) 4, 
The elliptic parameter v is now a fraction of the imaginary period, 
so that the Theta and D function required would have a complex argu- 
ment, and the tabular value is not available. 
The complete integral is expressible, however, by the function F(9) 
and K(); thus, from Legendre’s equation (m/’) F. E., t. I., p. 138, we 
find, with 
Ss) >aorSy As 83 
? Spy e 
(46) (2K) anf + Kn fB'= KB) — (KBP) 
where 
(47) sin?y = sn?fK/ = 81%, 
§; — 82 
cos?y = cn*fK’ = 7— *3, 
S,; — 8; 
A2x = dn?fK’ = ° — *3, 
5) — 89 
and the accent in E/(x) and F(x) implies the complementary 
modulus x’, 
So also for the other forms of the integral ; thus 
L/(—2z , 
(48) | lan . A pain See 
and four times this integral will give the expression for Q, the conical 
angle subtended by a circular or elliptic dise at a point, the equivalent 
of the magnetic potential for normal magnetisation. 
The same complete Elliptic Integral of the Third Kind is required 
for the determination of the apsidal angle in the Spherical Pendulum 
or of the symmetrical spinning top. 
