498] ON THE INVERSION OF A QUADRIC SURFACE. 71 
which is of course satisfied by 0 = 0. Moreover the derived equation 
_ vpj— ^ a ' 2 m 2 b' 2 n 2 c 2 
<W+Wf‘~(W+by ~<J+df 
is also satisfied by 0 = 0, so that this is a double root. The equation in fact is 
[6 2 d +0ol- (l 2 a' 4- m 2 b' + n 2 c')) (0 + a') (0 + b') (0 + c) 
4- {l 2 a 2 (0 + b') (0 + c) 4- m 2 b' 2 (0 + c) (0 + a') + n 2 c' 2 (0 + a') (0 4- 6')} = 0, 
or, expanding and dividing by 0 2 , this is 
d(0 + a')(0 + b')(0 + c') 
+ a [0 2 + 0(a +b' + c) + b'c + c'a' + ab'} 
— (l 2 a! 4- m 2 b' + n 2 c') (0 + a' + b' + c') 
4- l 2 a' 2 4- m 2 b’ 2 + n 2 c' 2 = 0, 
which gives the remaining three roots. 
If a' = 6' = c' the equation is 
(0 4- a' 4- a.) (0 + a') 2 = 0. 
I recall that we have 
7 7 , mn nl , Im , 
a, b, c, d, /=-7-, g = ~T> h =j> b m > n > 
a ' = a -Z, b' = b-^, d = c-%, ct— l 2 + m 2 + n 2 , 
so that the quadric surface is 
d (a!sc? + b'y 2 + c'z 2 ) + (lx + my 4- nz 4- dw) 2 = 0, 
and that, a lt ¡3 1) a l} b u c x denoting as before, the equation of the inverse surface 
(referred to a different origin) is 
r 4 = 4w 2 \cl-iX 2 4- ¡3{y 2 + 71 z 2 4- S 1 w 2 4- (cq# 4- bpy 4- Ciz)}.