If Sn be a rational integral function of x of the nth degree, and Sn-1 Sn-2… S1 S0 a series of such functions of the n — 1th, n—2th, &c., degrees, so related to Sn that, when any one of the whole series S0 S1…. Sn vanishes, the two on opposite sides have opposite signs, and farther Sn-1 and Sn have always opposite signs when x is just less than any real root of Sn = 0, then S0 S1… Sn-1 may be called a set of Sturmians to Sn. 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.