88 Mr Bromwich , Theorems on Matrices and Bilinear Forms. 
the expansion of \rE x —f {A^)\~ l in powers of £) = ?’— /(a); it is 
readily seen to be 
(pE, - FJ- 1 = + 1 F, + \ F t ' + . . . . . + - 0 Fr\ 
P P 2 P 3 pi 
n k 
rk 
where F (a) + 
/ t+1 («) + ...+ 
C,- 1 
(«- 0! 
and <7 will be the index of the first invariant-factor to base p. In 
order to determine q, we note that F x q = 0 and Fp~ x =)= 0. Hence 
we must have qlc ^ a, while (q — l)k<a, for F x q = 0 provided 
C x qk = 0 and C x a is known to be zero. If then we write a — kl-\-m, 
where m = 1, 2, ..., k, we have q = l -f 1 . Clearly l is the quotient 
and (m — 1) the remainder in dividing {a — 1) by k. 
To determine the number of invariant-factors which take the 
form p q , we have a theorem of Stickelberger’s 1 connecting this 
number with the rank of the form multiplied by l/p q in the 
expansion of \rE x — /’(H 1 )] -1 . This form is here F 1 q ~ 1 = F 1 l ; its 
rank is thus the same as that of G x kl , or is a — kl = m. Thus we 
have m invariant-factors p q . To determine the remaining in- 
variant-facfcors we have to apply an extension of Stickelberger’s 
theorem' 2 and calculate the rank of the coefficient of 1 jp q ~ x in 
our expansion; this rank will be (2??2 + the number of invariant- 
factors p q ~ l ). Now the rank in question is that of F x q ~ 2 — F x l ~ l 
which is that of C x k (i_1) ; or is a — k (l - 1 ) = m + k. Thus we have 
(k — m) invariant-factors each with index q — 1 = l. 
Further m (l + 1) + (Jc — m) l — kl + m = a, 
and so these k invariant-factors are all that correspond to the one 
(?• — a) a , of the original form ; which is verified by calculating the 
rank of the coefficient of 1 /p s+1 in ( pE x — Fi) -1 . 
We have then the theorem — 
If A be a bilinear form, one of whose invariant-factors is 
(r — a) a , and \{f (A) be a function of the form such that 
f («) = 0, f" (a) = (a) = 0, /* (a) + 0 
(while /(a) may or may not be zero), then f(A) will have k 
invariant-factors 3 corresponding to the one (r — a) a ; of these m 
1 Crelle’s Journal , Bd. 86 (1879), p. 42, Satz vn ; see also Muth’s Elementar- 
theiler, p. 135, Satz xvi; and also the paper quoted next. 
2 The proof given by the author, Proc. Lond. Math. Soc., vol. 32 (1900), p. 86, 
can be used to show that if 
(\^4 -B)~ 1 = Z 1 (\-c)~ e + Z 2 (\- c) -e+1 + ... +Z e (\-c)~ 1 + positive powers of (X-c), 
then the rank of Z r is [rw x + (r- 1) m 2 + ... +w r ], where m k denotes the number 
of invariant-factors of | - B | of the form (X - c) e ~ k+1 . 
3 It ought, perhaps, to be pointed out, that these invariant-factors do not 
necessarily form consecutive sets; as there may be others [r-/(a)] m from other 
invariant-factors of the original form. 
