88 Mr Bromwich, Theorems on Matrices and Bilinear Forms. 



the expansion of [rE 1 —f{A^j\~ l i n powers of p = r — f(a); it is 

 readily seen to be 



(pE, - F^ = -E 1 + \f i + \f-> + ... +- Q F^-\ 

 p p 1 p 3 pQ 



Qk fjk+l n <x-l 



where ft = ^f (a) + ^y,/** 1 (a)+...+ j^rjy/- 1 («), 



and ^ will be the index of the first invariant-factor to base p. In 

 order to determine q, we note that F-P = and F-^" 1 =j= 0. Hence 

 we must have qk 5 a, while (g — l)/c<a, for F^ = provided 

 G x qk = and Cj 01 is known to be zero. If then we write a = H + m, 

 where m = 1, 2, . . . , k, we have q = Z -f 1. Clearly / 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 pi, we have a theorem of Stickelberger's 1 connecting this 

 number with the rank of the form multiplied by \\pi in the 

 expansion of [rE 1 —f(A 1 )]- 1 . This form is here F 1 ^~ 1 =F 1 l ; its 

 rank is thus the same as that of C-f 1 , or is a — kl = m. Thus we 

 have m invariant-factors p q . To determine the remaining in- 

 variant-factors we have to apply an extension of Stickelberger's 

 theorem' 2 and calculate the rank of the coefficient of l/pi' 1 in 

 our expansion; this rank will be (2m + the number of invariant- 

 factors p 9-1 ). Now the rank in question is that of F^~ 2 ■= F^' 1 

 which is that of (7 X * (Z_1) ; or is a — k (I - 1) = m 4- k. Thus we have 

 (k — m) invariant-factors each with index q—1 — I, 



Further m (I + 1) + (k — m) I — kl + m = a, 



and so these k invariant-factors are all that correspond to the one 

 (r — a) a , of the original form ; which is verified by calculating the 

 rank of the coefficient of l/p s+1 in {pE 1 — i^) _1 . 

 We have then the theorem — 



If A be a bilinear form, one of whose invariant-factors is 

 (r — a) a , and if f(A) be a function of the form such that 



/' (P) = 0, f" (a) = 0, , . ., /*- (a) = 0, /* (a) =j= 



(while f(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 



(XA - B)~ l = 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 [rmj + (r-l) m 2 + ... +m r ], where m k denotes the number 

 of invariant-factors of | \A - B | of the form (X - c) e_fc+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—f(a)f +1 from other 

 invariant-factors of the original form. 



