354 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
, - , . vmwsin(pcos(p 
Jbrom (155.) we may deduce smr= ^ 7 = . 
u iwu c u i sina cosa 
we shall therefore have tanr secr=— — ^ ^ — 
(165.) 
MN 
It may easily be shown that tanr seer represents the portion of a tangent to a parabola 
intercepted between the point of contact and the perpendicular from the focus. 
dr 
(167.) 
TT «r<lT fl 
Hence tanr secr= 2 I — » — 1 - 
jeos^r JeosT 
Combining (164.), (166.) and (167.), and using the ordinarynotation of elliptic integrals, 
dr 
cosr’ 
or as 
we have therefore 
dr d sinT 
1 Cdr 
flf 
1 Idipl 
' C mn sinip coS(p"] 
- C 1 — ^2 sin^^ J ^ 
COST 1 — sinV ’ 
VninJcosT 
C to/iL _ 
[ mn sirup cosp“] 
L sin^p -1 
(168.) 
1 1 
1 1 
■ C mn sinp cosp”] 
. C 1 — sin^p J 
dp 
V mn ' 
1- 
P "C mn sinp cosp" 
L 'C 1 — sin^p ■ 
r' 
(169.) 
This is the expression given by Legendre, Traits des Fonctions EUiptiques, tom. i. 
p. 68. Written in the notation adopted in this paper, the formula would be 
Cdip 1 C dr 
mnj V'l v'^jeosr* 
(I/O.) 
XLI. We may express a and h, the semiaxes of the elliptic base of the cylinder, in 
terms of m and w, the conjugate parameters of the elliptic integrals in the preceding 
equations. From the equation of condition 7n-\-n—mn—t^, and (130.) we may elimi- 
nate and get 
[n — ’ k^ {n — mY ( /-•) 
Therefore 
b ^ f\ — n C(1 — 7i)(l — to) 'Cl — j 
a V 1 — TO (1 — to) 1 — TO 1 — TO 
Hence the ratio of the axes of the elliptic base of the cylinder is a function of the 
modulus and parameter. 
The ratio of the corresponding quantities in the case of the spherical ellipse may 
be derived from the equation 
or 
This ratio is therefore independent of the parameter. There is then an important 
difference in the two cases. In the one case, the ratio of the axes is independent of 
the parameter, and will continue invariable, while the parameter passes through every 
