DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 65 
dition of the conjugation of parameters in the circular form is 
(1 -f-rc)(l — m) — (\ — i 2 ). 
Now l+w^L l-ro=^, 
b 2 a 4 ar 
whence the proposition is manifest. 
An invariable relation exists between the parameter m and the modulus i ; for as 
a 4 — b 4 j . cd—b 2 m . 
m= — and i 2 — — ^-+* = 2, (477.) 
hence i being given, m also is given ; or the elliptic lemniscate can be represented by 
only one species of spherical ellipse, that in which the sum of the principal arcs is 
equal to two right angles. 
On the Hyperbolic Lemniscate. 
XCII. The equation of the lemniscate in this case is 
(x 2 -\-y 2 ) 2 =a 2 x 2 — b 2 y 2 . 
Following the steps indicated in (XC.), we find 
ds 2 a 4 cos 2 X + b 4 sin 2 X # 
d> 2 « 2 cos 2 X — b 2 sin 2 X ’ 
the limits of X are 0 and tan -1 y 
Assume sin ’ X = (478,) 
The limits of <p, corresponding to A=0 and X=tan -1 |? are <p=0, and <p=^- 
Substituting this value of sin 2 >. in the preceding equation, we shall find 
dk~ cosf (479.) 
From (478.) we may derive 
d\ a 2 b(a 2 + b 2 ) cosp 
dtp ~~ [a 2 b 2 + a 4 sin 2 <p + b 4 cos 2 <p] \/ a 2 + b 2 cos 2 <p" 
Multiplying the two latter equations together and reducing, we get 
eft f* dty 
,,,w 
When a—b, or when the lemniscate is that of Bernoulli, we get the well-known 
expression 
a C d<p 
MDCCCLIV, K 
