A SPECIAL CLASS OF STURMIANS. 165 
If we put p= —2, we get 
eee ap) (4p +279") . 
Each of these leads to the well-known condition for the reality of the roots 
of the cubic. 
7. It follows at once from (2) that, if two roots of the equation be equal, 
then S,(z) vanishes identically, and 8,_,(z), S,_,(z), . . . S,(z), form a Stur- 
mian series for the roots all supposed single. If three roots be equal to one 
another, or if two pairs be equal, then §,(z) and S,_,(z) vanish identically, and 
the rest form a Sturmian series for all the roots supposed single ; and so on. 
The present class of Sturmians present therefore an instructive contrast to the 
ordinary series obtained by the method of the greatest common measure. 
VOL. XXX. PART I. 25 
