APPLICABLE TO ELLIPTIC AND ULTEA-ELLIPTIC EUNCTIONS. 
419 
according to whether the upper or lower sign be taken. Now, because § is an ^th root 
2kir 
of unity, *=2 cos-^, and the sum of the pair reduces itself, for v'N, to 
4rN 
2rN 
(r2 + N) - cos (r^ - N) sin^ + N cos^ 
2?’N 
kTT 
2?’N 
kir 
N + sin^— (r2-N) r^-cos^^ (r^-N) 
(3.) 
For N * we have the simpler forms, 
1 4r 
2 /* 
Zk= 
* / o SAtT , a i o • ok'JT ,, o k'lr 
(r^ + N) — cos — T- (r®— N) sm^ — + N cos^ — 
i z z 
2 r 
2r 
N + sin^ y (r2 - N) * - cos^ y {r^ - N) 
(4.) 
All that remains is to integrate these terms, and sum them. 
6 . Our grouping the terms in pairs has limited the value of k to range from iio 
when ^ is odd. There is an odd term which, however, presents no difficulty, being 
simply j in the case of -v/N, and ^ in the case of N"^. When i is even, k is limited to 
-j- ^ 
range from 1 to \i—l, and the odd term becomes — y- in the case of a/N, and ■ - 
in the case of N“^. It is important to bear in mind that the term just mentioned is an 
odd term, and therefore not affected with the coefficient 2 , which appears in the terms 
composed of pairs corresponding to imaginary roots, 
7. The value of ^, which I consider to be most useful for general purposes, is ?‘= 8 : in 
9*^ -j- /CTT 
this case the odd term becomes — 5 — or — , and the other valnes of — are three in 
number, viz, 22° 30', 45°, 67° 30'. With proper precautions ^=8 will almost always 
give seven or more figures correct. 
8 . If we now give infinite values to k and i and pass from the summation to the defi- 
nite integral, we have ^putting 
2rd\ 2 r d<^ 
VNh , r^-N . 
14 — sit 
==^-*=i i 
r® + (N —7-^) cos^ Xtt' 
N 
and since 
j: 
d^ 
this is an identical equation, as it ought to be. 
9. This use of approximants, therefore, is simply the application of the method of 
3 L 2 
