169 
508] IN PARTICULAR THOSE OF A QUADRIC SURFACE, 
or expanding and reducing, this is 
{Y + (\-C)X}> 
+ Y(- 2 / a + 4X(7-40 2 ) 
+ X (2pX-^v-2fiC) 
+ fxr — 4 vC = 0, 
or say 
/jr — 4<vC 
+ (2/jX — 4v — 2{xC) (p + q) 
+ (— 2/j, 4- 4X(7 — 40 2 ) pq 
+ (X - C) 2 (p 2 + q 2 + 2pq) 
+ 2 (X-C)pq (p + q) 
+ p 2 q 2 = 0, 
viz. this, containing the constant C, is the general integral of the differential equation 
§V_a.^L-o 
VP VQ - ’ 
where 
P = (a+p)(b+p)(c+p), =v + pp + \p 2 +p 3 , 
Q =(a + q)(b + q)(c + q), = v + pq + X# 2 + q 3 . 
21. The constant G is connected with the parameter cr, which originally served 
to determine the right line, by the equation 
1 _ 2 (/3 — a) (X — (7) — /36 + aa 
a + a~y c-(X-C) ’ 
or, what is the same thing, 
1 _ 2 2 c 2 — a 2 — b 2 — C (2c — a — b) 
a a b — c C—a — b 
Reverting to the equation between p, q, cf>, I remark that if </> be therein con 
sidered as variable, we have the differential equation 
\JQdp + yT dq + Vd? d(j> = 0, 
where P, Q have the foregoing values 
P = - 8 a 2 /3 2 ((f) 2 -ry 2 )(ci + b-c- (f>)(a+p)(b+ p) (c +p), &c.; 
and where, if the integral equation be written in the form 
L+2M(f) + N<f) 2 =0, 
then we have <3> = M 2 — NL, viz. we thus find 
= 16a 2 /3 2 (a +p) (b +p) (c +p) (a + q)(b + q) (c + q). 
C. VIII. 
22