DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 63 
Assume 
a 
tanX= ^ tan<p 
(471.) 
Making in the last equation the substitutions suggested by this transformation, we 
get, after some reductions, 
, ds 
b-7~= 
a 2 cos 2 <p + 1 2 sin 2 <p 
dtp' 
1+ (V^) sinS<> v/ 
(472.) 
Let 
a 2 —b 2 a 2 — b 2 .„ 
-W~= n ’ 
(4/3.) 
and the last equation will become 
a 2 + b 2 { 
? 4 
,.C ‘‘f 
~ b J 
1 [I + w sin 2 <p] /y/ 1 — i 2 sin 2 <p 
J V 1 — i 2 sin 2 <p 
. • (474.) 
On the plane ellipse as a base, let a vertical cylinder be erected, and from the 
centre of this ellipse let a sphere with a radius = v' a 2 -\-b 2 be described. This sphere 
will cut the elliptic cylinder in a spherical conic section. The expression for an arc 
of a spherical conic section measured from the extremity of the minor arc is given by 
the equation 
s cos/3 ( 
d<p cosa cos /3( 
r 4 
R cosa sina’ 
1 [1 + tan 2 s sm 2 ip] -/l - sin 2 >] sin 2 <p sina J 
| a/ 1 — sinSj sin 2 p 
See Theory of Elliptic Integrals, p. 27. 
Now in this case a, and (3 being the principal angles of the concentric cone whose 
base is the spherical conic section, 
• 2 "90 b 2 
sma! =SqT°’ 81n 0=?+j* 
therefore cos 2 a=^-^p, cos 2 /3 = 
7T 
Hence a+|3 = g> 
or the sum of the principal angles of this cone is equal to two right angles, or the 
cone is its own supplemental cone. From these equations we may infer that 
coSjS \Za 2 -\-b 2 cosa cos/3 b 
cosasina” l) J sina 
. 9 sin 2 a — sin 2 /3 a 2 —b 2 . „ sina — sin 2 0 a 2 — b 2 
tarns = s — -= — 75—, sin 2 7?=— — =■ -= - „ - . 
cos 2 « b 2 5 0111 ' sina a 2 
Making these substitutions in the preceding equation, we get 
