Theory and applications of finite groups, by G.A. Miller, H. F. Blichfeldt [and] L. E. Dickson.

208

HIERhMITIAN INVARIANT[

[CH. IXI

by a change of variables of the following type:
yi = plixi,
Y2 = p21X1 + p22X2,
ys=PslX1+Ps2X2+. P s&Xs2
Yn=Pta1Xi+Pn2X2~ ~PnsXs+... ~PnnXn.
Proof. Arranging J according to x7, and x we have
J=Jn =h qnnxn-n+n -l+XnXn-,+X,
where Xn-1 represents a linear function of XI,, x,~- 1.
The coefficient q,,n is real and positive, since it is the value
of J obtained by putting x=l, Xn-1=Xn2=... =XI=0.
Accordingly,
+ vnn = ~vAnn,
and we may write
= (v~zXn X-1 -(v/-1un7~ Xn' X - 1Xn
ynyn +Jn -; say,
where
yn-i
Vqnn
and is therefore a linear function of x.,..., XI.
The function J1,- fulfils the conditions of an Hermitian
form in n -i variables xn i,..., xi, as it is of the required
type and is real and positive for any set of values allotted to
these n-i variables except 0,..., 0. For, it is the value
of J. obtained by putting x.= -X -jql/q. Hence, we may
arrange J.1 according to x,-i and i,-j and proceed as above.
We find
hn- I =yn - 1n - 1 +y-iJ -2,