78 Mr Bromwich, Theorems on Matrices and Bilinear Forms. 
we have B = B' and BB' = B 2 = Xc r x r y r , 
and on substitution 
(B~'RA) (B-'RA)' = E = (ASB~ 1 )' (. ASB -') 
or the two forms B~ l RA, ASB~' are orthogonal. Now the product 
of two orthogonal forms is itself orthogonal, so as R, S are or- 
thogonal we have 
P = B~'(RAS), Q = (RAS)B-' 
as two orthogonal forms. 
Hence RA (, SP ~ *) = B = (Q~'R) AS, 
and SP~\ Q~'R are both orthogonal, so the original problem has 
been solved. But in one case it is found that we do not need to 
calculate P , Q. To prove this we have that 
Q = BPB- 1 
is an orthogonal form and thus Q = (Q') _1 . 
Or, since B' = B, we have 
BPB-' = ( B-'P'B )-' = B-'PB, 
for P is orthogonal and so ( P / )~ 1 = P. 
Hence B 2 P = PB 2 and B 2 = ^c r x r y r , 
and if all the coefficients c r are different, this equation can 
only hold if P is of the form 2 d r x r y r , where the d’ s may be any 
arbitraries. Now P is orthogonal and so we must have 
d r 2 = 1. (> = 1, 2, ..., n) 
In this case 
RAS=BP = 2(±cJ)x r y r , 
and the form has been again reduced by the two orthogonal sub- 
stitutions R, S. Obviously a sufficient condition (though not 
necessary) is that all the roots of \AA'-\E |=0 should be 
different. In this case we have say 
AA' = 2c r X r Y r , A' A = tc r B r U r , 
and then A = 2 (± c r %) X r Yi r 
which is Sylvester’s rule. 
But if the roots of j AA — \E j = 0 are not all different it may 
be necessary to modify one or other set of variables 1 by a further 
1 That is, either the X’s or the H’s (it is immaterial which); the fact that the 
two substitutions R, S do not necessarily reduce A seems not to have been noticed 
before. 
