DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 383 
If we divide this equation by 2\/z, the first member will be the differential of 
. Integrating this equation. 
z 
ifSt- - 
(326.) 
1 ^“2^ 
Assume w = r sin-0, then "tan^0 ^ dw=2? sin0 cos0d0 (327.) 
Hence 
* n^'^x 
d0 
ltan^0 V' 1 — sin‘^0‘ 
(328.) 
We must now integrate this expression, 
cl0 
tan^0 sin^0 
r dd 
1 sin^0 v/i — i 
=fil 
d0 
sin^0 v' I— sin^0 
cot0 
- 2^ sin^0 
f d0 
J 1 — sin‘‘^0 
ll’ cos^0d0 
J 
r i^cos®0d0 
J{1 — 2^ siu^0)^ 
20)1 
. d0 22sin0cos0 
1(1 
adding these equations, 
d0 22sin0cos0 
v' 1 — 22 sin20 ~J -/ ( 1 _ ^2 sin20)f 
_ r _ (1 dg 
J 1 — 2^ sin20 ^ "^1(1 — 22sin20)3 
— id9\/ 1 — sin^y ; 
sin20 
(329.) 
tan20''/l — 22sia20 V'l— 22sin20 V^l— 22sin20 is , 
t9\/ 1—P sin20+Jd0\/ 1 —P si 0^0 (299.). 
d?2 
' sin20 
(330.) 
c 
c 
r d0 
Jtan20 V 1 — i 
• d?2 
We have next to compute the value of the integral 
Now 
JnV^x J ''Z 1 — sin20 el 
Substituting these values of the integrals in (326.), 
If we now substitute this value of J in the equation given in (173.) for a 
quadrant of the logarithmic ellipse, namely, 
2 V' 1 — S [2ra — 2‘2— 
1 — n o- 
r2 df ii^-n)r2 d^5 C 2 y- 
J. nVT+~J. Vi+i 
. 272 — 2® — 
Since - =(1— ^'^sin^0)— cot^0, we shall obtain the resulting equation, 
3 D 
MDCCCLII. 
