174 ME. C. W. MEEEIFIELD OX THE COMPAEISOX OE HTPEEBOLIC AECS. 
Equation (8.) leads to a formula for the direct reduction of the logarithmic integi-al of 
the third kind, whose parameter is negative and greater than unity. It is the exact 
analogue of Legendee’s formula for the reduction of the same integral where the para- 
meter is negative and less than unity, pp. 153, 154 of his third volume on Elliptic- 
Functions. The reduction is of some importance, because on it depends the possibihty 
of tabulating those functions, which would otherwise require a table of treble entry, too 
cumbrous to attempt. 
Let CO, and be two amplitudes, such that, for the common modulus 0, we have 
Fw, = F?) + Fc4 . . . .] 
(a). 
. . . .J 
We must have simultaneously 
=—cos^^ tan a tamp tansy, . 
Hpi -1- Ha — H 0 J 2 =—cos^0 tan a tan (p tan ■ 
and also, putting for shortness for '/(!+ cos^^tan^pi), 
taiKpcosaSa — tan « cos 
\ — cos^ tan^ « tan^ 9 
_ tan (p cos + tan « cos (p 8 a 
ansyg-— ^ cos^ tan® « tan^ 
Let us next consider the function 
dwa 
a = V.,-V«,= H., 
If we regard a as constant, we obtain from equations {a.). 
dw^ df du), 
/\cu^ A<p Aojj’ 
whence 
■J 
{i'h 
(e). 
Now formulse (b.) and (c.) give 
Hsya— Hsyi = 2Ha+ COS® 0 tan a tan <p (tan co 2 -\- tan cj,), 
and 
whence 
2 tan <p cos ctAa 
tan CO.) -j- tan co, « ’ 
^ ' * 1 — cos 0 . tan^ « . tan-^ <p 
^ Cd<p[^ cos® 0 sin a. tan® ] 
*^”^1— cos® 0 tan®«tan®(pj’ 
an a.^ 
and, after a few reductions, we find 
1 /tt cos® 0 tan aX-o. , cos® 0 tan 
in=^Ha 
or, transposing, 
r ^ 
Jl — (1 + cos® 0 tan® «) sin® <p A<p cos® 0 tan a 
No constant is needed, since Vip is an even function of (p 
dp . 
A« 
(1 + cos® 0 tan® a) sin® p Ap 
{ ^(Vco.2 — Vicfi) — Ha . Fee } . 
