364 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
angle which a tangent to this element makes with the plane of the base, v in this 
equation is analogous to r in the last section. 
In the above expression, the negative sign is used as the curve is convex towards 
the origin. 
Now cos^X— sin% and tany= 
d.z 
dx 
\/(r:T+(ST 
We must substitute for these 
differentials, their values given in (223.), and introduce the value of <p assumed in 
. , 9 (A + C)2ACcos2(8 
(227.), whence ‘'= F[A+ C (235-) 
• ^ A[A + Ccos2;p] • 
But (231.) gives 
whence p secydX 
dx 
V" A + C.ak cos(p 
d<p \/B [A + C cos^ip] sin^ip ’ 
a^k cos^<pd<p 
t ij ”1 ^ ----- 
1 — sin2<pj Vl-i2sm2<p 
We must now determine the value of the second integral in (234.), namely, 
secydX, 
(236.) 
(237.) 
since cos^ 
%-y‘AaX g seccd;,^ - 
’ dx-' (a^cos^X— i^sin^)2 
Now we may derive from (223.) tanw: 
{a^ + y^) sinX cosX 
(238.) 
(239.) 
^(a^cos^X — 6^sin®X)^* 
Differentiating this expression, then multiplying by secy, and integrating, we obtain 
^ ^ cos^X + sin->X] secydx 
^Jcos^-t" +* )J („»cos»X-i*sin«A)S 
Comparing this expression with (238.), and introducing into (234.) the values found 
in (237.) and (240.), we obtain 
k Jcos^o 
making 
, A + B 
^“A + C’ 
since 
cos^ipdtp 
-v/B (A 4- C) J [1 — m sin^<p ] ^ — sin®<p ' 
C 
m= 
A + C’ 
• (241.) 
. (242.) 
*’=BlATcj> ”=7=B (243.) 
and we shall have m and n connected by the equation of condition, defined in (1.), 
m+w — mn~^. 
The three parameters /, m, n, and the modulus i are connected by the equations 
In—T?^ m+w— m»=^^ (244.) 
