39] ON THE DIAMETRAL PLANES OF A SURFACE OF THE SECOND ORDER. 257 
or reducing 
© = [F (G 2 - H 2 ) -GH {G- B)} m? ' 
+ {G(H 2 -F 2 )-HF(A-C)}y* 
+ [H (.F 2 - G 2 ) - PG (P - 4)} 
+ {G(A-B)(B-C) + FH(A+B-2(7) + G (P 2 + G 2 - 2tf 2 )}^ 
+ {H {B - G) (G - A) + GF (B + G - 2A) + H (G 2 + H* - 2P 2 )} za? 
+ {F(G — A) (A—B) + GH (C+ A — 2B) + F (H + F 2 — 2G 2 )} xy 2 
+ {H(B- G) (C-A) + FG (G+A - 2B) + H(H*+ H - 2G 2 )} tfz 
+ {F(C-A)(A-B) + GH(A + B-2G) + F{F* + G 2 -2H)}z*x 
+ [G{A — B) (B-C) +HF(B + C-2A) + G(G 2 + n-2F>)}x 2 y 
-{(A-B) (B-C) (C - A) + (B - C) F 2 + (C - A) G 2 + (A- B) H 2 } xyz. 
In the case of curves of the second order, the result is much more simple ; we have 
© = Ax + Hy, x — 0, 
Hx+By, y 
© =H (y 2 — x 2 ) + (A — B) xy = 0, 
for the equation of the two diameters. 
The above formulae may be applied to the question of finding the diametral planes 
of the cone circumscribed about a given surface of the second order, (or of the lines 
bisecting the angles made by two tangents of a curve of the second order). Considering 
the latter question first : if 
i+g-i-o 
a 2 b- 
be the equation of the curve, and a, /3 the coordinates of the point of intersection of 
the two tangents, the equation of the pair of tangents is 
or making the point of intersection the origin, 
(ßx — ay) 2 — (b 2 x 2 + (Gy 2 ) — 0 ; 
i.e. 
whence A = ft 2 -b 2 , B = a 2 — a 2 , H= — a/3, and the equation to the lines bisecting the 
angles formed by the tangents is 
aß (x 2 — y 2 ) — {a 2 — ß 2 — (a 2 — b 2 )} xy = 0, 
which is the same for all confocal ellipses; whence the known theorem, 
“ If there be two confocal ellipses, and tangents be drawn to the second from any 
point P of the first, the tangent and normal of the first conic at the point P, bisect 
the angles formed by the two tangents in question.” 
C. 
33