134 Proceedings of Royal Society of Edinburgh. [sess. 
in Stern’s Theorie der Kettenbriiche , the fourth section of which is 
given up to the consideration of such rules (Crellds Journ ., x. pp. 
4-7). 
The other observation is to the effect that 
“ every progression of terms constructed in conformity with 
the equation 
u n = a n u n _ i b n u n _ 2 -b C' n u n _ g db • • . . 
may he represented as an ascending series of principal coaxal 
determinants to a common matrix. Thus if each term in 
such progression is to he made a linear function of the three 
preceding terms, it will he representable hy means of the 
matrix 
A B C" 0 0 
1 A' B" C'" 0 
0 1 A" B'" C"" 
0 0 1 A'" B"" 
0 0 0 1 A"" 
indefinitely continued, which gives the terms 
1, A, AA'-B, A A' A" - BA" - AB" + C" , . . . .” 
This exhausts the paper so far as determinants are concerned : 
the results announced in it, one can readily own, were such as 
fairly to entitle the enthusiastic author to express his belief that 
“ the introduction of the method of determinants into the algorithm 
of continued fractions cannot fail to have an important hearing 
upon the future treatment and development of the theory of 
numbers.” 
Spottiswoode, W. (1853, August).* 
[Elementary theorems relating to determinants. Second edition, 
rewritten and much enlarged hy the author. Oe/Ze’s 
Journ., li. (1856) pp. 209-271, 328-381.] 
Save the utilisation of the fact that the denominator of any 
convergent of the continued fraction 
a. + 
a 0 + 
a 9 + 
* This is the author’s date at the end of the paper (p. 381). The first two 
parts of the volume, however, are dated 1855, and the remaining two 1856. 
