755] 
ON A MATRIX AND A FUNCTION CONNECTED THEREWITH. 
257 
is independent of x, and equal to either of these equal quantities; and if from these 
two values of u we eliminate X,, we obtain for u the quadric equation 
that is, 
cu 2 — (a — d) u — b = 0, 
cm+ b 
cu + d ’ 
as is, in fact, obvious from the consideration that n being indefinitely large the nth and 
(ft+l)th functions must be equal to each other. In the latter case, as \ n is indefinitely 
small, we have the like formulae, and we obtain for u the same quadric equation: 
the two values of u are however not the same, but (as is easily shown) their product 
is = — b -r c; u is therefore the other root of the quadric equation. Hence, as n 
increases, the function cf) n x continually approximates to one or the other of the roots 
of this quadric equation. The equation has equal roots if (a — d) 2 -1- 46c = 0, which is 
1—26 
the relation existing in the above-mentioned special case; and here u = (a — d), = ,, 
AG CL — Ct 
which result is also given by the formulae of the special case on writing therein n = oo. 
C. XI. 
33