A MEMOIR ON THE AUTOMORPHIC LINEAR TRANSFORMATION 
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of substitution for the nearer variables, the theorem will be verified if the result is 
equal to the matrix XI of the bipartite; that is, we ought to have 
n (il - T) -1 (i2 + - T)(il + T)" 1 0 = 0, 
or what is the same thing, 
0(0 - T) -1 (0 + T) 0- J (0 - T)(0 + T)- 1 0 = O ; 
this is successively reducible to 
(O + TOO-^O - T) = (O - T) 0- J (0 + T), 
0- ] (0 + TOO-^O - T) = 0- J (0 -T)0- 1 (0 + T), 
(1 + 0" 1 T)(1 - O^T) = (1 - 0~ 1 T)(1 + 0-*T), 
which is a mere identity, and the theorem is thus shown to be true. 
12. It is to be observed that, in the general theorem, the transformations or matrices 
of substitution for the two sets of variables respectively are not identical, but it may 
be required that this shall be so. Consider first the case where the matrix O is 
symmetrical, the necessary condition is that the matrix T shall be skew symmetrical ; 
in fact we have then 
tr. 0 = 0, tr. T = — T, 
and the transformations become 
(«, y, *) = (O-KO-T)(0 + T)-10$*„ y n Z/ ), 
(x, y, z) = (O-KO - T)(0 + T^Oftx,, y /5 z,), 
which are identical. We may in this case suppose that the two sets of variables 
become equal, and we have then the theorem for the automorphic linear transformation 
of the ordinary quadric 
(O##, y, zf, 
viz. T being a skew symmetrical matrix, if 
(«, y, ^) = (il->(il-T)(n + T)-‘fi$*„ y„ *,), 
then 
(0$®, y, «) 9 = (0$® / , y /} z)\ 
13. Next, if the matrix O be skew symmetrical, the condition is that the matrix 
T shall be symmetrical; we have in this case tr. O = — O, tr. T = T, and the four factors 
in the matrix of substitution for (x, y, z) are respectively — O -1 , — (O —T), — (O + T) _1 
and — O, and such matrix of substitution becomes therefore, as before, identical with 
that for (x, y, z); we have therefore the following theorem for the automorphic linear 
transformation of a skew symmetrical bipartite