520] 
ON THE CENTRO-SURFACE OF AN ELLIPSOID. 
317 
imaginary is of course attended to. The new results suitably modified would be 
applicable to the theory treated from the second point of view; but I do not on the 
present occasion attempt so to present them. 
The Ellipsoid; Parameters £, rj, &c. Art. Nos. 1—6. 
1. The position of a point (X, Y, Z) on the ellipsoid 
X 2 Y 2 Z?_ 
a 2 b 2 c 2 
may be determined by means of the parameters, or elliptic coordinates, £, i); viz. these 
are such that we have 
X 2 Y 2 Z 2 _ 
a 2 + £ + 6 2 + f + c 2 4- £” ’ 
X 2 Z*_ = 
a 2 +1]^ b 2 + c 2 + r/ ’ 
or, what is the same thing, £, 7) are the roots of the quadric equation 
X 2 Y 2 Z 2 _ 
a 2 4- v b 2 + v c 2 + v 
(In its actual form this is a cubic equation, but there is a root v = 0, which is 
to be thrown out, and the quadric equation is thus 
v 2 
+ v (a 2 + b 2 +c 2 — X 2 — Y 2 — Z 2 ) 
+ {b 2 c 2 + c 2 a 2 + a 2 b 2 — (b 2 4- c 2 ) X 2 — (c 2 + a 2 ) Y 2 — (a 2 + b 2 ) Z-] = 0, 
or putting 
P = a 2 + b 2 + c 2 , 
Q = b 2 c 2 4- c 2 a 2 + a 2 b 2 , 
R = a 2 b 2 c 2 , 
the equation is 
v 2 + v(P-X 2 - Y 2 -Z 2 ) + Q-(b 2 + c 2 )X 2 -(c 2 + a 2 ) Y 2 - (a 2 4- b 2 ) Z 2 = 0.) 
2. It is convenient to write throughout 
b 2 —c 2 = a, 
c 2 - a 2 = (3, 
a 2 — b 2 = y, 
(whence a 4 /3 4- y = 0).