DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 67 
Substituting- these values in (46.) the expression for an arc of a spherical ellipse 
with a positive parameter, and writing s for the arc, we get 
g 2 -6 2 _ 
G 3 
dtp 
b V a 2 + b 2 \ 
, 1+ (^) sin2? ]v / 
z> 2 
g 2 + Z> 2 
sin 2 <p 
b 3 p 
dtp 
G V G 2 +6 2 l 
/* «* + 6* sln!? 
Comparing this with (480.), we find 
V-6 2 \_ b 3 
s— 
s = 
a V g 2 + b 
V' 
v' a 2 — b 2 r. 
dp 
(484.) 
b 2 
a* + 4 9 
sin 2 p 
-j- v' a 1 — b 2 r, 
or the difference between an arc of a hyperbolic lemniscate and an arc of a spherical 
ellipse may be expressed by an integral of the first order, together with a circular arc. 
When a=b , the radius of the sphere is infinite, the sphere becomes a plane, so that it 
is not possible to express an arc of a spherical ellipse by the common lemniscate. 
Case II. Let b>a. 
In this case the arc of the hyperbolic lemniscate may be expressed by an arc of a 
logarithmic ellipse of a particular species, or one whose parameter and modulus are 
connected by the relation given in (481.). 
Resuming the expression in (480.) for the arc of the hyperbolic lemniscate. 
s= 
G 3 ^ 
bVW+ T 2 ! 
dp 
A 2 — G S 
1 2 ~ 
sin 2 <p 
V' 
b 2 
b 2 + d 
, sin 2 <p 
Let 
then as 
b 2 — g 2 b 2 .„ 
W+'a-— 1 5 
•9 ^ 
m-\-n—mn—r, 
(485.) 
Let A and B be the semiaxes of the base of the elliptic cylinder, k the parameter 
of the paraboloid whose intersection with the cylinder gives the logarithmic ellipse. 
Assume for the principal semimajor axis of the elliptic base 
A= vV+F. (486.) 
In (17L) vve found the following relations between A, B, k, rn, n, 
A 2 mn{\—ri) B 2 mn{\ — to ) 
k 2 ( n — to ) 2 5 k 2 ( n — to ) 2 5 
and as we assume A= v' a 2 +b 2 , we get, substituting for m and n their values in terms 
of a and b, the semiaxes of the hyperbola 
b 2 , , a 2 b 2 + g 4 — b 4 
