DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 363 
Differentiating" (222.), we get 
fdt/y 
a%'^ cos^A /i 
d.sry + b^)^a‘^b‘^ sin^A cos^A 
VdX/ (fl-cos^A — 
Ua/ — 
(a^ cos^A — sin^A)^’ C 
dx) A:^(a^ cos^A — sin^A)'^ 
Hence a%'’- {a^co^^x-b’^^m^xf ' ' 
Let this radical be put=\/V. 
Assume V=(A+B sin®?i.)(C — B sin“?i)=AC+B(C— A) sin^X— B^ sin% 
hence AC=a®^^ C — A=a^+6^ — P. . . . 
Let us now assume simp such, that 
AC sin®(p 
sm X= 
AB + BC cos^cp’ 
then 
A I p • 2 ^— + p B C(A + C) cos ^(p 
A+Bsm?i— C-BsinX— a^Ccos2(P ’ 
(223.) 
(224.) 
(225.) 
(226.) 
(227.) 
and 
., 2 „„ 2 . . 2 „-, 2 . ^2 («"+^'")ACsin"<p. 
a cosX b sin A— a B(A + Ccos2(p) ’ 
or as 
«^+&2=B, AC=ff^A" C+F=A+B, 
we get 
cos‘^X—b'^ sin~A== T 1 — T~^sinVl. 
A + Cco.s^(pL A + C 
Hence 
^ dL V AC . [A + C cos^ip] cos(p 
. (228 ) 
a^b'^ dA a'*(A + C) [1 — / sin'’'^(pj 
Making 
j A + B 
^ A + C 
, . . (229.) 
Differentiating 
, . . „ AC sin^a 
the equation sin’A= ab + BCcosV 
(230.) 
we get 
or as 
we get, finally. 
dx «A: v^A + C cosip 
'/B[A+Ccos2p]/^l-^(^^^jsin2<p 
dT dTdx C(A + B) 
d^ = dXd^’ * -B(A + C)’ • 
T b'^ C cos^(pd<p 
k a/B(A + C)J [1 — /sin^(p]^ ^/l — e^sin^ip' 
(231.) 
. (232.) 
. (233.) 
XLVI. We may develope another formula for the rectification of an arc of the 
logarithmic hyperbola. 
Assuming the principles established in Sect. XXXVIII., we may put 
T= 
secydX— 
secyd?t, 
(234.) 
In this formula p is the perpendicular from the axis of the hyperbolic cylinder let 
fall on a tangent plane to it, passing through the element of the curve ; and v is the 
