DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 59 
If vve write m for 
k — k x ( r‘ 2 —kk x 
, and i 2 for -y- 1 
Now as 
we get from these equations 
. sin 2 « — sin 2 /3 . 
i 2 =— — • o , and m — e — 
sin « ’ 
r — k x 
sin 2 « — sin 2 /3 
sin 2 «cos 2 /3 5 
„ kir — k-,) k,(r — k) 
tan a=— 7 - — 11, tan — jri, 
r(r—k) > " r(r—k x y 
(463.) 
(464.) 
whence 
— — tan/3 . _ {r—k)i 
^r 2 -kk x 
V k{r — k x ) 
Making these substitutions, (462.) will become 
s — ✓ r 2 — kk x 
tan/3 
tana 
sin 4[wS 
d<$ 
(465.) 
sin 2 <p] \/l — sinrj simp * 
Now, as we have shown in (16.), this expression denotes an arc of the spherical 
ellipse whose principal angles are given by the equations (464.), and whose radius is 
\Zr*—kk x . Hence if a sphere be described whose radius is not r, but ^r 2 —kk x , the 
length of the curve, the intersection of the sphere (r) with the paraboloid (kk x ) will 
be equivalent to that of a spherical ellipse described on the sphere whose radius is 
v 7 r 2 —kk x . 
When r=k, k being greater than k x , (459.) becomes 
r]o 
\ / k(k — k l ) or s=2k(k— k x )9. 
Hence s is an arc of a circle. That such ought to be the case is manifest, for in this 
v- 
k — k x 
is the 
case the sphere intersects the paraboloid in its circular sections, and ^ ^ 
cosine of the angle which the plane of the circular section of the paraboloid makes 
with its axis. 
We have shown in the first part of this paper that the curves of intersection of 
concentric surfaces of the second order may be rectified by elliptic integrals. When 
the intersecting surfaces are not concentric, the rectification of the curve of inter- 
section may be reduced to the integration of an expression which may be called an 
hyperelliptic integral. 
The general expression for the length of an arc of this curve will be an integral of 
the form 
/«« 4 + / 3 a , 3 + 7^ 2 + 8 ^+g 
J V ax* + bx s + cx 2 + ex +/ * 
When the surfaces are symmetrically placed and have a common plane of contact, 
the above expression may be reduced to 
olx a + /3a; 2 + <yx + 8 
ax 3 + bx 2 + cx + e 
This is also an hyperelliptic integral. 
