495] 
ON THE ENVELOPE OF A CERTAIN QUADRIC SURFACE. 
49 
and thence a- = ^ , &c., and substituting these values y disappears and we have 
p \f(x 2 + Xa 2 ) + q ^{y- + X/3 2 ) + r ^(z- + Xy-) + s \/(w- + \fr) = 0, 
a *P l ffg , r f r , _ n 
V(« 2 + Xa 2 ) \% 2 + X/3-) V(^ 2 + X7 2 ) VO 2 + XS 2 ) 
from which X is to be eliminated; the second equation is here the derived function 
of the first in regard to X, so that rationalising the first equation, the result is, as 
will be shown, of the form (*][X, 1) 4 = 0, and the result is obtained by equating to 
zero the discriminant of the quartic function. 
Denoting for shortness the first equation by 
A + B + C + D = 0, 
the rationalised form is 
(A* + B i + C i + D i - 2A-B- - 2A-C- - 2A-D- - 2B-CJ- - 2B-D- - 2C-DJ - UAB 2 C 2 B- = 0, 
which is of the form 
- (21 + 223X + (5X 2 ) 2 + (a, b, c, d, e$l, X) 4 = 0, 
where 
2f = p i B ... — 2p-q-x-y-..., 
23 = p 4 aV... — p 2 q 2 (ary- + /3hf-)..., 
(5 = p i a i ... — 2p-q 2 a-fi 2 ..., 
a = 8. x 2 y 2 z 2 w 2 , 
4b = 8. oi 2 y 2 z 2 w 2 + ..., 
6c = 8. a 2 j3 2 z 2 w 2 + ..., 
4d = 8 . a 2 /3 2r fw 2 + ..., 
e = 8 . a 2 fi-y 2 &. 
Writing T. J' for the two invariants we find without difficulty 
F = I - fP + A 2 , 
J' — J — Q + |AP — -^A 3 , 
where 
I = ae — 4bd -f* 3c 2 , 
J = ace — ad 2 — b 2 e — c 3 + 2bcd, 
A = 21(5 — 23 2 , 
p = a(5 2 - + 2c (21(5 + 233-) - 4d2l23 + e2l 2 , 
Q = (ce — d>) № + (ae + 2bd - 3c=). j (216 + 233=) + (ac - b=) 6= 
-2 (ad-be)236 - 2(be-cd) «SB. 
C. VIII. 
7