APPLICABLE TO ELLIPTIC AifD ULTEA-ELLIPTIC FUNCTIONS. 
423 
(6) If we make ^=tan we obtain 
(1 — sin^ 0 . sin^ ip}~* d(p=2{l — 2 cos 2 6 . dt. 
Taking r=l — the terms which we have to integrate are of the form 
, kw 
1(1 — #^)^ + 4cos^9 . cos^ — . 
i 
ki: 
Putting q^=l — cos^ 6 . cos® — , we have 
log. 
The same expression serves for the integral 
2dt 
i/(l + 2 cos 2 5 . + 
kir 
if we put §'®=1 — sin®^. cos®—. 
19. It will be observed that the first four cases, and the sixth, depend upon a radical 
k-n 
of the form a/( 1 — sin® A . sin® a/), where u is restricted to the selected values of -j. 
Assuming the modulus sin A not to vary, it would therefore in general be better to begin 
by computing the radical for the selected values. I have computed, and I append to 
this paper, a Table of this radical, the selected values of -j being 22° 30', 45°, and 67° 30', 
while A ranges by whole degrees from 1° to 90° inclusive. Every entry but the last in 
the 2nd, 3rd, and 4th columns of the Table was computed by myself in duplicate with 
Veg.As ten-figure logarithms, by the help of two or more of the following formulae, some 
of which are from Legendre. 
20. Putting A for \/(l — sin® A . sin®iy), 
(1) Make sin A . sin iy=sin M ; then A = cos M, 
log sin M=log sin A-j-log sin u, log A=log cos M ; or else 
(2) Make tan A . cosiy=tanM ; then A=cos A . sec M, 
log tan M=log tan A+log cos <y, log A=log cos A-j-ar. co. log cos M. 
Moreover, let L be the tabular angle nearest to the angle M : it is not necessary to 
obtain the value of M : so that we have simultaneously, 
log sin M=log sin L+s, 
log tan Mr=log tan L+^, 
log cos M=log cos L+c ; 
