130 
ON THE SIMULTANEOUS TRANSFORMATION OF TWO 
[188 
Comparing these with the relations between (x, y, z) and (x 1} y 1} z x ), we see that 
(£> V> y> z ) = (£i> Vii y\, ■S'iX 
and multiplying the first of the relations between two quadrics by an indeterminate 
quantity A, and adding it to the second, we have 
(\a + A, y, zf= (Acq + A x , ..yx u y u zj) 2 . 
We have thus a linear function and a quadric transformed into functions of the same 
form by means of the linear substitutions, and any invariant of the system will remain 
unaltered to a factor pres, such factor being a power of the determinant of sub 
stitution. The invariants are, 1° the discriminant of the quadric; 2° the reciprocant, 
considered not as a contravariant of the quadric, but as an invariant of the system. 
And if we write 
iT = Disc. (\a + A,...^x, y, zf, 
(21, 23, (£, ft, ©, y, £) 2 = Recip. (Aa + A,...Qx, y, z) 2 , 
then K 1} &c. being the analogous expressions for the transformed functions, and the 
determinant of substitution being represented by II, we have 
K, = n *K, 
(*!,-][&, VI, ^ = №(51,...^, v , m 
and substituting for £, y lf £ their values in terms of £, 77, £ the last equation breaks 
up into six equations, and we have 
K x = Id 2 A, 
(2li,.. a', a") 2 = №21, 
<&.-№ /S', /3") (y, y', y") = n 2 3, 
which is the system obtained in a somewhat different manner in my former paper. 
Putting /1 =g x = h x = F x = 6r 2 = H x = 0, and writing also (which is no additional loss of 
generality) a 1 = b 1 = c x = 1, the formulae become 
(a, b, c, f, g, h '§x, y, z)- = (1, 1, 1 '§x 2 , yr, 
(A, B, G, F, G, H\x, y, zf = (A u B 1} C&wf, yi 2 , z x 2 ), 
viz. there are two given quadrics which are to be by the same linear substitution 
transformed, one of them into the form x? + y? + z? and the other into the form 
AlX-? + B^i 2 + C^j 2 , where A lt B x , C x have to be determined. The solution is contained 
in the following system of formulae, viz. 
(A x + A) (i?j 4- X) (G x + A) = IT 2 Disc. (Aa + A,...),