356 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
If we now introduce the relation given in (130.) |= ^ 
substitution 
2V1 — , , /2n—i^—rfi\C d<p /i®— wNTdip , T, /y \ 
;r= -**'».+ (-^—)jN^+(-^)J;^+JdfN/i (1-3.) 
If we now suppose m=n, or or 2n—i^—rf=0, the last equation 
will become 
In this case 
( 1 - 6 .) 
O: 
tanip ^ I 
1 +^' tan‘^<p' (^7/-) 
This is the expression for the length of an arc of a logarithmic ellipse, the intersec- 
tion of a cylinder, now become circular, with a paraboloid whose semiparameter A’=0 ; 
therefore the dimensions of the paraboloid being indefinitely diminished in magni- 
tude, this intersection of a finite circular cylinder by a paraboloid indefinitely atte- 
nuated, must take place at an infinite altitude. We naturally should suppose that 
the section of a cylinder which indefinitely approaches in its limit to a circular cylinder, 
by a paraboloid of revolution, would be a circle ; yet the fact is not so. The inter- 
section of these surfaces, instead of being a circle, is a logarithmic ellipse, whose 
rectification may be effected by an elliptic integral of the second order, as we shall 
now proceed to show. 
In the first place let us conceive the paraboloid as of definite magnitude, and the 
cylinder to be elliptical ; its semiaxes as before being a and b. Then as a and h are 
the ordinates of a parabola, at the points where the elliptic cylinder meets the para- 
boloid, at its greatest and least distances from the axis of the surfaces, we shall have 
a^=:2hz’, h^=2ltz" (178.) 
Hence — }p=2lt{z' — z”). Let z' — z"—h, then h is the thickness or height of that 
portion of the cylinder within which the logarithmic ellipse is contained. 
Now (171 •) gives 
and we have also 
a 
b^= 
k^mn 
n — m 
2h = 
kmn 
n—m 
hence 
" 2 Vl-m 
n—m 
Now when n=m, a—b^ A=0, while we get for h 
a n 
2 V\—n 
(179.) 
We thus arrive at this most remarkable result, that though the cylinder changes from 
elliptic to circular, while the parameter of the paraboloid approximates to its limiting 
value 0, yet the thickness of the zone, that is h, does not also indefinitely diminish, but 
assumes the limiting value given above. 
Now if we cut this circular cylinder, the radius of whose base is a, by a plane 
