7 5 
728] 
A THEOREM IN ELLIPTIC FUNCTIONS. 
or, what is the same thing, 
— /3yx 2 = a+ p.a+q.a + r, 
— 7 ay 2 — b + p .b + q ,b + r, 
— afiz 2 = c +p .c + q.c+r, 
where a, /3, y denote b — c, c—a, a — b respectively; then, treating r as a constant, 
the coordinates x, y, z will belong to a point on the ellipsoid 
, 2/ a 
a + r b + r 
+ 
c + r 
1, 
and the differential equation of the right lines upon this surface is 
dp _ dq 
^a+p .b+p .c + p 'da + q.b + q.c + q 
Take x 0 , y 0 , z 0 the coordinates of a point on the surface, and p 0 , q 0 the corresponding- 
values of p, q, so that 
- /3yx 0 2 = a + p 0 • a + q 0 . a + r, 
~ 7 a 2/o 2 = b +p 0 . b + q 0 . b + r, 
- a(3z 0 2 = c + p 0 ■ c + q 0 . c + r, 
then the equation of the tangent plane at the point (x 0 , y 0 , z 0 ) is 
flagp + W o + zz ±_ = i 
a + r b + r c + r 
or, substituting for x 2 , x 0 2 , &c., their values, we have 
Br/XX o / 0 
— — = Vd + p . a + q . a + p 0 . a + q 0 , &c., 
a + r 
and consequently the equation of the tangent plane is 
as/a+p .a + q.a +p 0 . a + q 0 + /3 \/b + p .b + q .b +p 0 .b + q 0 
+ y^/c+p.c+q.c+p 0 .c + q 0 = -CL/3y, 
the equation of a plane intersecting the ellipsoid in a pair of lines; hence this 
equation (containing in appearance the two arbitrary constants p 0 and q n ) is the integral 
of the proposed differential equation. 
Writing 
sn 2 u = A (a + p), cn 2 u = B (b + p), dn 2 u = C(c +p), 
the values of A, B, G, k are determined; and, assuming for q, p 0 , q 0 the like forms 
with the arguments v, u 0 , v 0 , the differential equation becomes du=dv, having the 
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