DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 335 
Differentiating this expression with respect to and [Jtj, 
(1+7) cos^ju. +y sin^fx. 
sin^^J/ djx cos^ju. sin^jx 
We have also 
tan^^/: 
(1+7)^ sin^/x cos^j«. 
(cos^ft.— y sin^jix.)^ 
Whence, after some reductions, sin^^'^ 
Multiplying this expression by j s-iid reducing 
'/i- 
-?V 
2, ’ 
+y 
(70.) 
(71.) 
(72.) 
(73.) 
Multiplying; together the left-hand members of the equations (70.), (72.) and (73.), 
and also the right-hand members together, we get, after some obvious reductions, and 
integrating, 
('+4-Vf^^ ( 74 .) 
dtl/ 
sin^jx ' 
This is the well-known relation between two elliptic integrals of the first order whose 
moduli are i and or in the common notation, whose moduli are c and 
j tan/x 
XXVI. Let T be the arc whose tangent is , , 
then 
tan‘^r cos/x V 1 — sin^;x _ 
cos^jx — sin'^jX ’ 
and combining (71.) and (73.), we shall find 
t®nt|/ p sirijx coS|X -v^l — sin^/x 
cosV~,;^ sin'^jx 
Dividing (75.) by (76.), the result becomes tan2r: 
JL 
1+7 
tanti/ 
sin^4' 
(75.) 
(76.) 
(77.) 
We are thus enabled to express r, the portion of the tangent arc between the point 
of contact and the foot of the perpendicular arc on it, in terms of 4' instead of |M/. 
If we introduce this value of r into (62.) and combine with it the relations esta- 
blished in (74.), the resulting equation will become 
dfj/ p d\t/ 
dfi/ c dvt/ 
J [' “ - (t^)' J \/* - 
+7 
^-(W) 
tan' 
_27 
1+7 
T tan^/ 
MDCCCLII. 
1+JJ 
2 X 
( 78 .) 
