If Snbe a rational integral function ofxof thenthdegree, and Sn-1Sn-2... S1S0a series of such functions of then— 1th,n—2th, &c., degrees, so related to Snthat, when any one of the whole series S0S1.... Snvanishes, the two on opposite sides have opposite signs, and farther Sn-1and Snhave always opposite signs whenxis just less than any real root of Sn= 0, then S0S1... Sn-1may be called a set of Sturmians to Sn. It is obvious that the problem of finding such a set of functions admit of an

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... nite 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.