755] ON A MATRIX AND A FUNCTION CONNECTED THEREWITH. 
255 
and, if instead of these we consider the combinations a n x + b n and c n x + d n , we then 
obtain 
Sti)” 1 {(Vl+1 - !) (ax + b ) + (\ n - X) {- dx + 6)}, 
„ {(X n+1 — 1) (cx + d) + (\ n — X)( cx — a)}\ 
a n x + b n = - 
c n x d n == „ 
and in dividing the first of these by the second, the exterior factor disappears. 
It is to be remarked that, if n = 0, the formulae become as they should do a 0 x + b 0 = x, 
c 0 x + d 0 = 1; and if n = 1, they become a x x+ b x = ax + b, c x x+ d x = cx + d. 
If X m — 1 = 0, where m, the least exponent for which this equation is satisfied, is 
for the moment taken to be greater than 2, the terms in { } are 
(X — 1) {ax + b) + (1 — X) (— dx + b), 
and 
(X — 1) {cx + d) + (1 — X) ( cx — a) ; 
viz. these are (X — l){a-\-d)x, and (X —l)(a + d), or if for (X—l)(a + c?) we write 
viz. we have here 
or the function is periodic of the with order. Writing for shortness 'b = — , s being 
any integer not = 0, and prime to n, we have X = cos 2^ + i sin 2S-, hence 
1 + X = 2 cos (cos ^f + i sin S-), 
ILilM =4cos 2 ^-; consequently, in order to the function being periodic of the nth 
order, the relation between the coefficients is 
The formula extends to the case m = 2, viz. cos £ (stt) = 0, or the condition is 
ci + d = 0. But here X + 1=0, and the case requires to be separately verified. Recurring 
to the original expression for M 2 , we see that, for a + d = 0, this becomes 
a 2 + be, 0 
0 , d 2 + be 
, = (a 2 + be) 1, 0 
0, 1 
that is, 
a 2 x + b 2 
c. 2 x + d 2 
or the result is thus verified.