350 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
We also find for the criterion of sphericity -a. 
sm^ 
P-Q 
/P + Q\ /P-Ci\ 
cos 
( 141 .) 
As p is the altitude of a triangle whose sides are a, p, q, 
a^k^=(a+p+q)(p+q — a)(a-{-q—p){a 
XXXVIII. In the preceding investigations the element 
of the curve has been taken as a side of a limiting recti- 
linear polygon inscribed within it. We may however 
effect the rectification of the curve, starting from other 
elementary principles. Let APB be the plane base of the 
elliptic cylinder, and let a series of normal planes PPW 
rffTsr'vv' be drawn to the cylinder, indefinitely near to each 
other, and parallel to its axis. We may conceive of every 
element Pzn- of this plane ellipse between the normal 
planes as the projection of the corresponding element W 
of the logarithmic ellipse. Let r be the inclination of the 
element d2 of the logarithmic ellipse to the correspond- 
ing element d^ of the plane ellipse. We shall have, dX 
being the elementary angle between the planes PPW and 
+?-?)• 
Fig. 14. 
TJSZj'vv', 
dS 
ds 
d\ — ®^^’'dA- 
Now (31.) gives 
and therefore 
TIT IT „ , , 7, . , I d^» (a^ — — J-sin‘*x) 
In the plane ellipse cosrX4-h^ simX, whence ^ — • 
r x- 1 j clA'= (a^cos^A + o-sm^A)? 
We have now to express cost in terms of X. 
From (119.) combined with (120.) we may derive 
sec r _j_ sin^0 + b cos^ 6) 
V V V 1) 
Eliminating - between the equations tanX=T 2 and -=-tan0, we shall hav( 
X u X X CL 
( 142 .) 
( 143 .) 
( 144 .) 
( 1 - 15 .) 
tanX=^tan0 (H6.) 
If we eliminate tan0 by the help of this equation from (145.), we shall obtain 
l^{c? cos'A -f sin^A) 
COS r= 
6^— F] sia^A— sin^A" 
( 147 .) 
