the Function X 2 in the Application of Sturm's Tlieorem. 455 



France, and is now universally adopted, in the Continental 

 colleges, in place of the laborious method of Lagrange. 



It is presumed that the small volume which I have recently 

 published upon the theory and solution of equations, as also 

 Mr. Spiller's elegant translation of Sturm's memoir*, will 

 render any detailed account of the theorem itself unnecessary 

 in this place. My object in the present communication is 

 merely to offer a suggestion, for the purpose of facilitating a 

 little its practical application. 



The principal operation which Sturm's theorem involves, 

 is that of finding the greatest common measure between the 

 polynomial X, forming the first member of the proposed equa- 

 tion, and its derived function X r The series of remainders 

 or successive quotients at which we arrive in the course of 

 this operation, require to have their several signs changed as 

 they arise; and they then form, together with the original 

 polynomials X, X 19 a series of functions 



X, Xj , X 2 , Xg, ... 



from which every requisite information respecting the number 

 and situation of the real roots of the equation X = may be 

 readily evolved. 



Now, instead of taking the trouble of actually performing 

 the division of X by X 15 in order to obtain X 2 , the remainder 

 due to that division after having changed signs, we may find 

 X 2 much more easily by the following rule, viz. 



(A.) Having arranged the terms of X x so that they may be 

 severally under those involving like powers of x in X, proceed 

 thus : multiply the third term of X by twice the coefficient of 

 the first term of X, , the next term by three times that coeffi- 

 cient, the next by four times, and so on. From these several 

 results subtract those which arise from multiplying the terms 

 immediately under them by the second coefficient in X : the 

 remainders, taken with contrary signs, will form X 2 . 



(B.) When the second term ot X is zero, X 2 is obtained 

 by simply multiplying the terms of X, commencing at the 

 third, by —2, —3, —4, &c. respectively. 



Note. — Instead of using, as the rule directs, the coefficient 

 of the first term of X, , and the coefficient of the second term 

 of X, for multipliers, we may employ any two factors which 

 will make these coefficients equal and of the same sign as the 

 first in X r 



An example or two, selected from the volume on equations 



* A review of this work will be found in our last number, p. 384. 



