fakation) 
VI.—On a Special Class of Sturmians. By Professor CurystTAt. 
(Read 20th June 1881.) 
IfS,, be a rational integral function of x of the n™ degree, and §,,_18,_»...8, 8, 
a series of such functions of the n—1", n—2", &c., degrees, so related to S 
n 
that, when any one of the whole series 8) 8, ... 8, vanishes, the two on 
opposite sides have opposite signs, and farther S,_, and 8, have always opposite 
signs when x is just less than any real root of 8,=0, then S, 8,...8,_, may 
be called a set of Sturmians to S,. It is obvious that the problem of finding 
such a set of functions admit of an infinite number of solutions. The first 
discovery of such a set was made by Sturm, and the researches of SYLVESTER, 
Hermite, and others have shown how other solutions of the problem may be 
obtained. 
It occurred to me while working at some physical questions that the 
properties of symmetrical determinants would furnish us with the means of 
constructing a particular class of Sturmians. I thought when I found the 
result that it was new, but a little research led me to a paper by J OACHIMSTHAL 
(CRELLE’s Journal, Bd. xlviii. p. 386), where the very same series is given. 
The method by which I independently arrived at the result is so simple and so 
different from that of JoACHIMSTHAL that I have thought it worth while to lay 
it before the Society. 
1. Let 
A= 41 Wy2...- Ain 
21 M2 » « » Aon 
Ani On2 eos Onn 
be a symmetrical determinant, so that @,,=d,,, &c.; and let us call the deter- 
minant formed by deleting the first row and first column, the first two rows 
and the first two columns, and so on, its first, second, &c., principal minors. 
Then we have the well-known proposition that, if any principal minor 
vanish, the next higher and the next lower have opposite signs. This is easily 
VOL. XXX, PART I. 2A 
