DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 321 
Assume 
2 cos^X 2 Zi^sin^X 
^ a- cos^X + sin^X’ ^ cos^X + sin^X 
a-(A^ — a^) cos^X + Zi^(A^— ^^)sin^X 
^ a^ cos^X + sin^X 
Differentiating these expressions. 
/dr\® a^S'^sin^X a'^S'^cos^X 
\dxy [a^ cos^X + sin^x] ^ ’ y dx / [c? cos^X + b'^ sin^x] ® ’ 
and as xAx-\-ydy-\-zAz=0, ;> . 
a'‘6‘*(a^— 5^)2 sin^X cos^X 
\dX/ [a^cos®X + &^ sin^X]^[a^(A:^— a^)cos^X + ^»®(A^ — 5^) sin^X]"^ 
Substituting these expressions in (20.), we find 
/d(r\ ^ cfib'^ [a^(P — a®) cos^X + — b^) sin^X + (a^— 6^)^sin^Xcos^X] 
\dX/ A^[a^cos^X + 6^sin®X]^[a^(A:^ — a^)cos^X + Z»^(F— Z»^)sin^X] ' 
The numerator of this expression may be resolved into the factors 
[jf cos^X+J^ sin^X] [{k^—a^) cos^X+(F— 6^) sin^A], 
and the equation may now be written 
Assume 
do- 
dx 
a^b’^ V (yt^ — a-)cos^X+ {k^—b^) sin^X 
A:[a^cos^X + Z»^sin^X] -v/ a^(A:® — a^)cos^X + 6^(^^ — 6^)sin^X 
( 21 .) 
( 22 .) 
(23.) 
(24.) 
(25.) 
Hence 
dx_ v/(/t^-a^)(F-Z>^) 
dvl/ — a^) sin^4/ + {k^ — b^) cos^4' 
(24.) may now be transformed into 
do- 
do- dx 
a^^^/{k^-a^){k^-b^) 
dxd4' k[a^[k'^ — i^)cos^\I/ + 5^(^® — a^)sin®4/] 'v/a®cos^\I/ + 6®sin^4' ^ ^ 
If we imagine a concentric cone to pass through the mutual intersection of the 
cylinder and the sphere, we shall have 
a=A:sina, Z> = A: sin/3, 
a^-b^ 
sm ;j=- 
e^= 
tan^a — tan®j3 k^{a^ — b'^) 
Whence (26.) may be transformed into 
tan/3 
tana 
tan^a 
dv|/ 
■a2(F-62)_ 
e® sin^4'] V \ — sin^)) sin^vl/ J ’ 
(27.) 
( 28 .) 
an expression identically the same with (16.). 
The angle -v// in this expression is identical with <p in (16.). 
.p 2 12 cos^X + b'^ sin^X a^ + b‘^ tan^X _ 
^ cos^X + b'^ sin^X a^-\-b'^ tan^X ’ 
eliminating tanX by (25.), 
2 2 6d(A:^— 6 ^)cos^ 4'4-5"^(^^ — a^)sin®4' 
^ ^y a^{k^ — 6^) cos^\I/ + — «^)sin^4' 
2 T 2 
