A SPECIAL CLASS OF STURMIANS. 1638 
if a, a... . ad be the roots of 
B+ pe i+... +p,=0. 
3. When z=a, 
= becomes (ayaz.. . An)?E(a, . . « An) (a1 —A2) » . « (1 — Gn) 
and 
Sn—1(%) becomes (agaz . . . an)’E(az. . . An)(ay—az) .. . (ay —aan). 
Hence, when v=a, 
Bol) / $,_1(2) =a? {(1—as) ras)» « (@h—a)}®. 
Now, if a, be real, and p be an even positive or negative integer, this ratio will 
be real and positive ; for a, a,... a, being by supposition the root of an equa- 
tion with real coefficients, for every imaginary in the series a, — a,, a, —a3...4,— On, 
there will occur a corresponding conjugate imaginary so that the product of them 
all will be real. 
It follows that S,_,(v) and S,(v) have opposite signs when 2 is just less than 
any real root of 
SZ) = 0, 
which is the second characteristic of the first two functions of a Sturmian 
series. 
The restriction as to p being even may be removed if positive and negative 
roots be considered separately; but for simplicity I shall suppose p to be 
always even. 
4, If we take the determinantal expression for S,, multiply each column 
by 2, and subtract the next following, leaving of course the last column 
unchanged, we get, denoting for brevity s,v—s,,, by (p), Spu:%—Sp4. by 
(p+1), &e., 
S,(”) =| (p) Ga Oe... Kp +n—-1) 
GE) (Oreo) = = En) 
@Gr2) . @E3prs)  .-. (p+n+1) 
| (p+n—1)(p+n)(p+n+l1) ... (p+2n—2) 
which it will be observed is a symmetrical determinant. S,_:(z), similarly trans- 
formed, becomes the first principal minor of this obtained by deleting the last row 
and the last column, and so on. Hence, by (1), S,(), S,-1 (7)... S,(z) S,(z), 
