328 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
we shall have 
tan^a=^ 
i^(l — m) ■ 
20 * 
tan j3= 
(53.) 
In order that these values of tana, tan|3 may be real, we must have m>i^ and ;a< 1. 
Let the parameter be positive. 
Now 
hence 
^ „ sin^« — sin^jS , . „ sin^« — sin'^S 
tan g=- 2 =n, and sim;?= — — =^^ 
sin^a 
tanV— tan"^=|^^^\ 
(54.) 
There is in this case no restriction on the magnitude of n. 
XIX. To determine the value of the expression 
(’■?)«/»*« 
fr 
df 
. — sin’^ipl 
L(1 +>i sin^f) V 1 • 
when n is infinite. 
As m — n-\-mn=i^, ov {\—m){\-\-n) = \ — i^T=.j^, 
when n is infinite, m=l. 
Resuming the expression given in (47.)> 
d<p 
-4=f- 
d<p 
J L(1 +?i sin^<p) v' 1 sin'^ipj V mnj V 1 — sin^(p 
we find that when n is infinite, a is a right angle. 
For 
— tan' 
r 
mn siDcp cosa~ 
L \/l- 
i- sin-if _ 
^ „ sin^a — sin^/3 , „ -a 
w = tan £=— — = 00 , therefore 
COS^a 
Now being the angle between the spherical radius vector drawn to the extremity 
of the arc, and the major principal arc, we have 
, tan®/3 , cos« , tan/3 sin/3 
tan^/^^^^ tan?., and tan^=^^ tanX, or Ian tan^. 
Hence is indefinitely less than (p, when n is infinite, or when a is a right angle. In 
this case therefore ff=0, and we get, when n is infinite, and <p not 0, 
d(p 
;]=i 
+ n sin^cp) \/ sin'^cp. 
We might have derived this theorem directly from (4/.), by the transformation 
n sin<p= tan^y. 
This is case I. in the Table, p. 316. 
Section III . — On the Spherical Parabola. 
XX. It remains now to exhibit a class of spherical conic sections whose rectifica- 
tion may be effected by elliptic integrals of the Jirst order. 
The curve which is the gnomonic projection of a plane parabola on the surface of 
a sphere, the focus being the pole, may be rectified by an elliptic integral of the first 
order. 
