DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 319 
arc of a curve, taking p instead of RS the independent variable. We may derive 
from (6.) the following expressions, 
) — sin^/3 ' 
•sin^/Sf 
. ,, sin^/3 fsin^a— sin^fl I sin^a fsin^p — sii 
“ sin^p l^sin^a— siu^/3j ’ ^ sm'^p [^sin^a— sii 
. . . (10.) 
Differentiating the former with respect to ■p' P? eliminating sin 4', cos 4 ; 
using for this purpose the relations established in (10.), we find 
d\t/ — sina sin/3 cosp 
dp sinp '/sin^a — sin^p -v/sin^p — sin^(3 
dvf/ . 
. (11.) 
Substituting this value of ^ in the general expression for the arc ; the resulting 
equation will become 
sinp cos^p — cos®« cos^/3 
)]’ 
v/ (sin^a — sin^p)(sin®p — sin^/3). 
an elliptic integral which may be reduced to the usual form by the following trans- 
formation : assume — 
2 sin^a cos^(p + sin®/3 sin^<p 
COS p cos'^f + tan'^3 sin'^ip’ 
(13.) 
TT 
The limits of integration are 0 and g. Differentiating this expression, and intro- 
ducing into (12.) the relations assumed in (13.), we obtain for the arc the following 
expression : — 
tan/3 . 
rr 
d^ 
5 
/tan^a— tan^/3') 
1 tan^a / 
sin^ip 
\/ 1- 
^sin^a — sin^/3'\ 
1 sin^a / 
sin^^ 
. . . . (14.) 
Let e be the eccentricity of the plane base of the cone, whose semiaxes are A and B, 
as in (V.), 
A^— tan^a — tan®^ 
A2 
tan^a 
as in (4,), 
(3.) gives 
. „ sin®« — sin®/3 
sim;?= r-s , 
and we derive from (2.) 
or grouping these results together, 
. „ sin®« — sin^/3 
sin^£= — ^ ^ ; 
cos^p 
tan®« — tan^/3 
tan^a 
= m 
. „ sin^« — sin®/3 
Sin ;?= ^-2 — 
( 15 .) 
. , sm^a— -sin^/3 
sin^g — 
cos®/3 
2 T 
■ n. 
JVIDCCCLII. 
