mmmmam 
[153 
y, this 
153] OF A BIPARTITE QUADRIC FUNCTION. 
8. But in like manner the equation 
(tr. il$E — x, H — y, Z - z) = — (tr. T][E, H, Z) 
501 
gives 
and we then obtain 
(tr. 0 + T£E, H, Z) = (tr. il][x, y, z), 
or 
(tr.il-T£E, H, Z) = (tr. iTftx,, y„ z,). 
9. In fact these equations give 
(tr. 2il#E, H, Z) = (tr. il][x + x /} y + y /} z + z), 
2(E, H, Z) = (x + x / , y + y„ z + z)-, 
and conversely, this equation, combined with either of the two equations, gives the other 
of them. We have then 
and thence 
(x , y , z) = ((tr.il) Htr.il + TftE, H, Z), 
(E, H, Z) = ((tr. O^T)" 1 tr. il$x„ y„ z), 
(x, y, z) = ((tr. il)“ 1 (tr. il + T)(tr. il - T)-»tr. il$x /} y„ z 7 ). 
10. Hence, recapitulating, we have the following theorem for the automorphic linear 
transformation of the bipartite 
(ii£®, y, y> z )> 
viz. T being an arbitrary matrix, if 
(®, V, z) = (n~'(Cl-T)(il + T)-^ilfe, y„ z), 
then 
(x, y, z) = ((tr. il)“ 1 (tr. il + T)(tr. il — T) -1 tr. il$ X/ , y„ z), 
(il][a;, y, z#x, y, z) = (il$>„ y„ zftx,, y„ z), 
which is the theorem in question. 
11. I have thought it worth while to preserve the foregoing investigation, but 
the most simple demonstration is the verification a 'posteriori by the actual substitution 
of the transformed values of (x, y, z), (x, y, z). To effect this, recollecting that in general 
tr. (A -1 ) = (tr. Z.) -1 and tr. ABGD = tr. D. tr. G. tr. B. tr.il, the transposed matrix of 
substitution for the further variables is 
iltil-rr^il + T) il- 1 ; 
and compounding this with the matrix il of the bipartite, and the matrix 
il" 1 (il - T)(il + T)" 1 il 
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