164 PROFESSOR CHRYSTAL ON 
the last being any positive constant, have the property that, when any one 
of the series vanishes, the next higher and the next lower have opposite 
signs. 
5. It has now been shown that 8,(z), S,_:(z), ...8,(z), S,(z) form a 
Sturmian series. By giving particular even values to p, we get of course an 
infinite number of such series. 
If it were desirable to employ these functions for the purposes of root 
discrimination, s,, s,-1, &c., could be calculated by Newron’s method, and by 
giving a proper negative value to p, the labour could be diminished by nearly 
half in the most general case. 
For example, if we take the cubic equation 
a+petq=0, 
and put p= —2, the Sturmian’s are 
S=—-|1 ¢@ 2¢@2@/,S=+12 « #/,8,==—|1 2 |,8=+1. 
G5 5.1 8p a een Sasa 
S_1 So Sy So S-1 So Sy Sj So 
So) Si Sada SSS Sh 
6. If we wish simply to find how many real roots there are, then we have 
simply to consider the signs of the coefficients of the highest powers of # in 
the Sturmians. This gives us the following theorem :— 
There are as many pairs of imaginary roots of the equation 
B+ pe + ... +p,=0 
as there are variations of sign in the series 
+1, 5, | S S41 |5| S Sp4i Spre |, &e. 
Sp+1 Sp+2 Sp+1 Sp+2 Sp+s 
Sp+2 Sp43 Sp44 
when p=0 this gives a well-known theorem (see Satmon, “ Higher Algebra,” 
p. 49). 
If we put p=0, the series for the cubic 
e+ pat+q=0, 
neglecting certain positive multipliers, is 
rh Ly +31, —6p, — (4p? + 279") . 
