366 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
or, if we substitute for the coefficients of this equation their values given in (246.), 
we shall have 
CcosVf + . . . (255.) 
sm<p cosip 
, ^ sin< 2 cos®vI 
Let i = P+P—- 2 li’, V—ne+P- 2 mi\ «>,= 
<t> =• 
M • (256-) 
Now the process given in XXX.VI. will enable us to develope the integrals 
and as follows 
and 
= m Jv/i 
The equations of condition ln=P, and m+n-nm = i\ give 
(Z-iT , (259.) 
m-\) 8sl n^cos(pi/I 
We have also, Since LM ' ‘ 
Making these substitutions, adding together (257.) and (258.), the coefficient of 
/d(p vanishes, and we shall have 
Ccos®(pcl<p 8 sirup cosip . 8(/— S 
M^vr lm 
but (255.) gives ' J v'T 
Combining this equation with the preceding, 
8 r dp 8'(/-i2)f dp , 8(Z-i^)rdp § sinp cosp / JzlLC^ 
— 5r‘jM7r+<(i=i)J';^+ rM -2 V /(i-i)Jcos>i 
(261.) 
7 
. 8'(7-i'^) (l-2")§ 
^ow — Hf}^ -\-i^ — ^ 2\2 5 cind 
{JZ-ff ’ l-v^ ’ m ~~ (/-I) 
In the last equation, substituting this value of I', and then dividing by 8, we get 
sinpcospA/I , (7— 
LM ^7(7-1) ] \Zl 
-*^)rdp _j7-7^)r^ _ 2 A/,^r-^. . . ( 262 .: 
7(7-T)JVT‘^ 7 Jlv/I (7-1)Jx\Ia/ 1 V 7(7-l)Jcos3o 
Now 2 j~ 3 „=tany secy+J^ and cos^y- 
as may be shown by combining (226.) with (235.). 
Hence siny=\/^|- 
LAI 
cos^p ’ 
(263.) 
( 264 .) 
