372 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
Introduce this value of j, and divide by 2, 
. (390.) 
Now when this equation is integrated between the limits <p=0, and ?)=sin~*.Y/^j o’’? 
taking the corresponding values, between r=0, and T=sin“'^Y:pj^, or between y=0, 
and y=|, T is infinite, and the arc of the asymptotic parabola is also infinite, 
but twice the difference A between those infinite quantities is finite. Let sin^ip,=y5 
'-^then . (291.) 
cos^’r 
Hence the difference between the two infinite arcs of the equilateral logarithmic 
hyperbola, and the corresponding infinite arcs of the asymptotic parabola, is equal to 
a right line + an arc of a plane parabola — an arc of a plane hyperbola. 
LI. On the logarithmic hyperbola, when 1=qo . Case XIL, p. 316. 
„ , , T C cos^ada 
Resume (233.), or yt — ^ c) J [I - ^ sin^-p] sin^ip ' 
Now as ln=i^, and as i is finite, while /=co , w=0. 
The equation of condition m-\-n—mn=i^, gives therefore m=T^. Equations (248.) 
and (249.) give a=0, h=k. 
And asx/B(A+C)=:^, we get m=i-=nh. 
hence 
cos^ipdip 
' [I — Z sin^ip] ^ V' I — sin^f 
Let / sin^<?i= sin^-^/, therefore /y/7cos<pd^=coS'47d-4/, [l — /sin^(p]^=cos^'4/j 
/ “^2 f sm^ 
^1— ?'^sin^<p=\/ 1— y sin^'4/=\/ 1— w sin^'xj^, and cos<p=^ 1 — — y-. 
Making these substitutions in the preceding equation, we get 
(a.) 
^=- 7 = r*^t, ^ . When /=oo, 7 = 0, w=0 ; hence T=A:(--^^, (292.1 
k V^cos^V ■v/l— rasm24; ’I ’ ’ Jcos^vp ^ 
or the logarithmic hyperbola in this case becomes a common parabola. 
As a=0, h=k, the hyperbolic cylinder becomes a vertical plane, at right angles to 
the transverse axis. 
Hence, comparing this result with (XIX.), we find that when the parameters are 
either +oo or — oo, the corresponding hyperconic section is a plane principal section 
of the generating surface, i. e. either a circle or a parabola. 
