153] 
OF A BIPARTITE QUADRIC FUNCTION. 
503 
when the transformations for the two sets of variables are identical, viz. T being any 
symmetrical matrix, if 
(*, y> 4=(n- 1 (ii-T)(ii+T)-^ / , y„ »,), 
(x, y, z) = (il-(il-T)(il + T)-^][x / , y„ Z/ ), 
then 
y> Z H X > y> z )= (!!$>„ y„ z^x, y„ Z/ ). 
14. Lastly, in the general case where the matrix i2 is anything whatever, the 
condition is 
il-^T = - (tr. il)" 1 tr. T 
for assuming this equation, then first 
XI- 1 (12 - T) = (tr. X2)~ 1 (tr. X2 + T), 
and in like manner 
+ T) = (tr. i2) _1 (tr. X2-T). 
But we have 
1 = (tr. i2)^(tr. f2^T)(tr. tr. i2, 
and therefore, secondly, 
(fl + T)“ 1 12 = (tr. 12 - T)" 1 tr. O. ; 
and thence 
il-^n - T)(i2 + T) -1 i2 = (tr. X2) -1 (tr. n + T)(tr. i2^T)-Hr. i2, 
or the two transformations are identical. 
15. To further develope this result, let X2 _1 be expressed as the sum of a 
symmetrical matrix Q 0 and a skew symmetrical matrix Q /} and let T be expressed in 
like manner as the sum of a symmetrical matrix T 0 and a skew symmetrical matrix 
T r We have then 
^ -1 = Qo + Q/> 
(tr. I2) -1 = tr. (X2 _1 ) = Qo-Q,, 
T =T 0 + T / , 
tr. T =T 0 -T /} 
and the condition, i2 _1 T = — (tr. i2) _1 tr. T, becomes 
(Qo + Q,)(T 0 + t,) = - (Q 0 - QX T 0 - T # ), 
Q,T 0 + QA-0, 
T o=-Qo~ 1 Q, r ,> 
that is, 
and we have