130 Proceedings of Royal Society of Edinburgh. [sess. 
the properties of the functions referred to.* Their connection 
with determinants, after the awakening of interest in the latter 
about 1841, was sure sooner or later to be detected : there is no 
evidence, however, of the discovery having been made before the 
year 1853. 
Sylvester, J. J. (1853, May 13). 
[On a remarkable modification of Sturm’s theorem. Philos. 
Mag. (4), v. pp. 446-457.] 
The mention of Sturm’s theorem in the title of a paper renders 
not improbable the occurrence therein of matter connected with 
continued fractions. Especially likely is this in the case of a 
writer like Sylvester when in a characteristic mood ; and, assuredly, 
the present communication is in structure, style, and originality 
redolent of its author. It must have been written in the white 
heat of discovery. The main part of it consists of six pages : 
this is followed by a “ Remark ” a page and a quarter long ; then 
comes a “ Postscript ” of three and a half pages ; and finally a 
small-page footnote as long as the “ Remark.” 
It is the postscript which particularly concerns us. It begins 
thus : — 
“ Suppose that we have any series of terms, u x , u 2 , u 3 , 
. . . , u n , where 
U\ — > u 2 = AjA 2 — 1 , u 3 = AjAgAg — Aj — A 3 , ... 
and in general 
U £ == i u ^_ 2 ) 
then u lt u 2 , u 5 , . . . , u n will be the successive principal 
coaxal determinants of a symmetrical matrix. Thus suppose 
n = 5 ; if we write down the matrix 
A x 1 0 0 0 
1 A 2 1 0 0 
0 1 A 3 1 0 
0 0 1 A 4 1 
0 0 0 1 a 5 
* The state of the theory in 1833 can best be gathered from Stern’s 
monograph published in vol. x. of Crelle's Journal. 
