672 Proceedings of Poyal Society of .Edinburgh. [sess. 
of importance because of the author’s peculiar view that the con- 
tinued fraction 
h, 
CL i + ^ 
a 2 + . 
K 
+ - 
a n 
is best expressed by means of two determinants of the form 
/A 
\/ <2^ 
The series obtained is, as might have been expected, that to 
which he would have been led by the use of the combinatorial 
“ rule ” more than once already referred to, and found fully 
enunciated in Stern (Crelle’s Journal , x. p. 6). It is indicated 
in substance by saying that the coefficient of x in A is the sum 
of all the products of p factors taken from a x 2 , a 2 ’ 2 , . . . . , , 
subject to the condition that no two factors with consecutive 
suffixes shall occur. Curiously enough, the finding of this sum is 
viewed as a problem with the following theorem for a solution : — 
“ Lehrsatz. — Der Coefficient der Potenz x n ~ 2p in der Deter- 
minanten-Entwickelung hat den Wertli 
K = n — 2p + l r=p — 2 
2, * a ^+2-l ' a l+2.2 * 1 * J a !+2r * M, 
k= 1 r — 0 
unter M die (p — r — 1) fache Summe 
% = n - 2p + 2r + 3 s. 2 = n- 2p + 2r + 3 + 2- 1 
jr 2 
Sj = k + 2r + 3 s. 2 = k + 2 r + 3 + 2*1 
s p _ r _ 3 = ra-2p + 2r+3 + 2(p - r - 4) 
2 ' 
Sp-r - 3 = « + 2?’ + 3 + 2 (p — r — 4 ) 
