258 ON THE DIAMETRAL PLANES OF A SURFACE OF THE SECOND ORDER. [39 
In the case of surfaces, the equation of the circumscribing cone referred to its 
vertex as origin, is 
whence 
X s V 2 Z‘ 
4- j— 
a a b 2 c 2 
a 2 b 2 c 2 
A = ß 2 c 2 + 7 2 6 2 — b 2 c 2 , 
B = 7 2 a 2 + a 2 c 2 — a 2 c 2 , 
(7 = a 2 5 2 + /3 2 a 2 — 6 2 a 2 , 
F = — a 2 ßy, 
G = — 6 2 7« , 
H = — c 2 a/3. 
Hence, omitting the factor 6 2 c 2 a 2 + c 2 a 2 /3 2 + a 2 6 2 y 2 — a 2 lrc 2 , we have 
& = a 2 — a 2 , 
a3=/0 2 -6 2 , 
© = ry2 - c 2 , 
$ = Py> 
(3r = 7 a, 
?^ = a/3; 
and the equation of the system of diametral planes becomes 
= 0 = a 2 /3y (c 2 — 6 2 ) P + /3 2 7a (a 2 — c 2 ) # 3 + y 2 ot/3 (b 2 — a 2 ) P 
+ 7a [a 2 (c 2 — 6 2 ) + /3 2 (6 2 + c 2 — 2a 2 ) — 7 2 (5 2 — a 2 ) -f (6 2 — a 2 ) (c 2 — 5 2 )} yz 1 
+ a/3 {— a 2 (c 2 — 6 2 ) + /3 2 (a 2 — c 2 ) + 7 2 (c 2 + a 2 — 2/> 2 j + (c 2 — b 2 ) (a 2 — c 2 /} zx 2 
+ y<x [a 2 (a 2 + b 2 — 2c 2 ) — /3 2 (a 2 — c 2 ) + y 2 (b 2 — a 2 ) + (a 2 — c 2 ) (5 2 — a 2 )] xy 2 
— a/8 {a 2 (c 2 — 6 2 ) — /3 2 (a 2 — c 2 ) — 7 2 (b 2 + c 2 — 2a 2 ) — (a 2 — c 2 ) (c 2 — 6 2 )} y 2 z 
— 187 {— a 2 (c 2 + a 2 — 26 2 ) + /3 2 (a 2 — c 2 ) — 7 2 (6 2 — a 2 ) — (i 2 — a 2 ) (a 2 — c 2 )} ¿sV 
— 7a {— a 2 (c 2 — 6 2 ) — /8 2 (a 2 + b 2 — 2c 2 ) 4- 7 s (6 2 — a 2 ) — (c 2 — 6 2 ) (b 2 — a 2 )} x 2 y 
4 {(a 2 — b 2 ) (b 2 — c 2 ) (c 2 — a 2 ) 4- 
(a 4 4- /S 2 7 2 ) (6 2 — c 2 ) — (/8 4 4- 7 2 a 2 ) (c 2 — a 2 ) — (7-* + a 2 /8 2 ) (a 2 — 6 2 ) 4- 
a 2 (b 2 — c 2 ) (2a 2 — b 2 — c 2 ) 4- /3 2 (c 2 — a 2 ) (26 2 — c 2 — a 2 ) + 7 2 (a 2 — b 2 ) (2c 2 — a 2 — 6 2 )} xyz ; 
and since this is a function of a 2 — b 2 , b 2 — c 2 , and c 2 — a 2 , the equation is the same 
for all confocal ellipsoids; whence the known theorem, “ The axes of the circumscribing 
cone having its vertex in a given point P, are tangents to the curves of intersection 
of the three surfaces, confocal with the given surface, which pass through the point P.”