245 
1915-16.] On the Theory of Continued-Fractions. 
where b (p) denotes the leading element in the 'power of the matrix 
u \ 
d 2 
0 
0 0 
0 0 
0 0 0 
0 0 0 
v n — 1 
0 
'n-l 
d„ 
(4) 
K I 
and where c x and are any two quantities whose product is a 1? c 2 and d 2 
are any two quantities whose product is a 2 , etc. 
It is generally most convenient either to take all the c’s to be unity, or else to take 
them equal to the d’s, so that the matrix becomes axisymmetric. 
For the convenience of readers who are not familiar with the theory of matrices, it 
may be explained that two matrices are multiplied together by the same rule as two 
determinants, the m th row of the first matrix being multiplied into the w th column of the 
second in order to give the element in the m th row and n th column of the product-matrix. 
In order to establish the theorem, consider the transformation from 
a set of variables (x 0 , x 1 , . . . x n ) to a set of variables (X 0 , X 1 , . . . X n ), 
defined by the equations 
X 0 = (6 + x)x 0 + cpc-^ 
Xj = dpc 0 + (&! + x)x l + c 2 x 2 
X 2 = + ifi + x)x 2 + c 3 x 3 
X n—1 - d n _ 1 x n _ 2 + (p n —i + x)x n _i + c n x n 
X n = d n x n _i + (b n + x)x n 
Suppose that when these equations are solved for the os’s we obtain 
x o — Poo*o + AiXi + . , 
- • + Pon^m ' 
x i = ft 0 X 0 + ftA + . 
• • + Pi n X n 
X n — finoX-Q 4“ /? n iXj + . . 
• “t h> n7 ix n _ 
( 6 ) 
From these last equations we see that /3 00 is the value of the ratio x 0 /X 0 
when X p X 2 , . . . X n are all zero : and therefore by (5) it is the value of 
cc 0 /X 0 derived from the equations 
X 0 = (6 + x)x 0 + cpx x 
0 = dpc Q + (&! + x)x x + c 2 x 2 
0 = d n x n _j + (b n + x)x n 
