DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 365 
I and n are reciprocal parameters, the reader will recollect, while m and n are con- 
jugate parameters. 
XLVII. It was shown in (226.), that C — A=a^+5^—A:^, A:^=A+B— C, 
and a-1f=AC, whence 
AC (A + B)(B-C) 
(A + B-C)2’ (A + B-C)2 
In order that these values of a and h may be real, we must have B>C, and A of the 
same sign with C, both positive, otherwise \/V in (225.) will be imaginary. As 
A + B 
l> 1 ; here the parameter I is greater than 1, v/hile m and n are each less 
than 1. 
We may express the semiaxes of the hyperbola, the base of the hyperbolic cylinder, 
in terms of the modulus i and the parameter /; for by the equations immediately 
preceding we may eliminate A, B and C in (243.). We thus find 
IP 
+ 2/2-2] 2 ’ ^2 — ^ j2 _ 2/^2] 2 ? 
(246.) 
r B a^ + b\_ /(/-I) 
theiefoie ^2— jp. —p^p_2ip 
(247.) 
We may express the semiaxes in terms of the conjugate parameters m and n, 
rPm{\—m) 
W m{\—n){n-\-m—mn) 
[n + m — 2/nn]^’ {n + m-~2mnY ' 
hence 
B a^-\-¥ 
m 
and \/B(A+C) = 
\/ 
mn 
(248.) 
(249.) 
k^ k^ (m-\-n — 2mn) V + n — 2mn) ’ * • • 
or we may express a and h more simply in terms of / and m. Eliminating n and ?, 
m(l — m) 5^ ^(^—1) 
we get 
(250.) 
k^ (/— ^2 ]/_,^]2 
Let c, be the eccentricity of the hyperbolic base of the cylinder, we shall easily 
discover the following equation between c^, i and /, analogous to (131.), 
{cf-\)ij^={l-iy (251.) 
Hence when i and I are given, c, may easily be found. 
XLVII I. If we equate together the values found for Y, the arc of the logarithmic 
hyperbola, in (233.) and (241.), we shall have 
'"f 
J[l — Zsii 
cos^ipdtp 
cos^ipdip 
I sin^^]^ V 1 —P sin^^ 
For brevity, put 
L=l — /sin®9, M= 1 — w sin®(p, N=l— wsin®(p, 1= 1 — sin^<p. 
The preceding equation may now be written 
msin2(p]2Vi_/2gin2c^— '^^(^ + ^)Jcos3o- ’ 
. (253.) 
Fcos^tsda / t, , . , jz: C do 
(- 254 .) 
