378 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
the same sphere by the reciprocal cylinder, or by the cylinder which gives a function 
having a reciprocal parameter, we shall find 
— e'^) 
But by the conditions of the question, as 
I • In *' 
ee' = ^, e = 
and 
i — sin®^’ 
sin^^d^ 
de' 
c ^ 
J 1 _^-2 , 
-v/ (e'^ — 2^) (1 — e'^) J 1 sin^0 ’ 
Cde' C f^mWde C 
J eisV 1375— Jv^l-ZsinS^— J^/i_/sinSS jd^^/l jsm'D. 
(310.) 
sin^0 
Substituting these values of the integrals in (309.) 
rr^_ 
ff = 
f'd(pv/l 
•'0 
J'^J L 
2 d® 
-v/l 
lyj 
(deyf 
+ constant. . 
(311.) 
We shall now show that the constant =0. 
When 0=0, e=l, and therefore e'=i. Since e'=i, and o- is a quadrant of the 
vanishing spherical ellipse whose principal arcs, a = 0, j3 = 0, we shall have s-=0. 
Hence also (cl0\/j=O, | ^=0 ; therefore the constant is 0. When 0=^ e'=l, 
and (309.) becomes 
*~in^ 'll”’*' ”11”’*’ ”11”^ 
".]+ Lt aj!j. 
^2 d0 
(312.) 
•jl 
or, in the common notation, ^=EiF^ + EyFj— F^F,, 
a formula already established in (308.). 
If we add together (307.) and (312.), we shall have, since \/ k 
- sin0 COS0 
V' 1 — siii'0’ 
d<p 
Vl 
(313.) 
Now <r=C-^)y;rnC-, ^rw 
\ m J [1 — ?/2 sin^ip] Vl — 2 ^ sin^<p \ ^/ / ^ ^’ [ 1 — 7?2y sin-^] V 1 — 
2‘*sin'*f’ 
in which mm =i^. 
Whence, as (h;r)'/’'‘'‘=(V^')'^ s/;, as we have shown in (113.), 
rr 
P : 
Ift fl — TOSi 
dip 
==^- + 
. cji n 
dip 
dip 
[1 — TOsin^ip] Vl— Psin'^ip \ V-, • 2 "1 /i .o " '0 Vl— 2^sin^ip 2 
^ ^ J ^1 ——siiPipJ V 1—2^ simp ^ 
-Ha. (314.) 
The reader will observe how very different are the geometrical origins of two alge- 
braical formulae apparently similar. In the logarithmic form of the elliptic integral, 
the formula for the comparison of elliptic integrals, with reciprocal parameters (one 
of which is greater, while the other is less than 1), resulted from putting in equation 
