370 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
bola ; the difFerence, however, between these infinite quantities is finite, and equal to 
, — — ^ — rG— S, integrated between the limits <p=0, and ^=sin“7“^. 
It is needless here to dwell on the analogy which this property bears to the finite 
difference between the infinite arc of the common hyperbola and its asymptot. When 
n—m, the above expression becomes illusory. We shall, however, in the next article 
find a remarkable value for the arc of the logarithmic hyperbola, when m=n. 
We may express the above formula somewhat more simply. 
As in (248.) 
^ iVnill—n) ^ Vmn{l—n) 
k m + n — 2mn’ k n — m 
bb^ 
mn{\ —n)m 
v' m [n—m)[n + m — 2mn)' 
The equation given in (285.) now becomes 
( 2 « 6 .) 
The ratio between the axes of the original hyperbolic cylinder, and of the derived 
elliptic cylinder, may easily be determined ; for 
— m) . , 
■m 
b^ I • „„ 
and -i=-. , (b.). 
i—n’ V ^ 
Let c, be the eccentricity of the hyperbolic base, and c that of the elliptic base, then 
rf{c ^ — l)=i^(l — c^). 
Comparing (a.) with (b.), x / n ^'=n / I |= 
This equation gives at once the ratio between the axes of the hyperbolic and elliptic 
cylinders. 
When the paraboloid becomes a plane, or when its parameter is infinite, w=0, 
S becomes an arc of a plane ellipse, is changed, into a rectilinear asymptot, and 
the expression in (286.) is now transformed into or the difference 
between the infinite branch of an hyperbola and its asymptot may be represented 
by an arc of a plane ellipse. 
L. On the rectification of the logarithmic hyperbola when the conjugate parameters 
are equal, or m—n. 
We have shown in XLII. that when m=n, the arc of the logarithmic ellipse is 
equivalent to an arc of a plane ellipse ; so when m—n, the arc of a logarithmic hyper- 
bola may be represented by an arc of a parabola, and an arc of a plane hyperbola. 
In (262.), if we make w=w, or l—\-\-j, n= \ —j, we shall have, writing N for M, 
and in (170.), if we make m=n, and M=N, 
O') 
I 
