of Sturm's Theor&fn. 4f 1 



We thus obtain (n) numbers /-tj, ju.^, . . . /x„, and bave only toibrm 

 a progression according to the well-known law f3diOXfa 



1, N,, N^, . . . N„, 



where Ni=/W', and in general Nj=//,j . Ni_i— -Nt_2« 



The number of arithmetical operations required by this method 

 (after the division part of the process which is common to the 

 two methods has been performed) will be 2/i multiplications and 

 2/1 additions or subtractions ; whereas if we deal with the residues 

 directly, the number of multiplications will be 



/ -. ^ -. • n.{n-\-\) 

 » + (/*~l)+... +1, «.e. y^^ 



(besides having to raise x to the wth power), and the same num- 

 ber of additions. The practical advantage, however, of this 

 method over the old method is not quite so great as it may at 

 first sight appear, in consequence of the quantities operated with 

 on applying it being larger numbers than those which have to 

 be used in the old method. 



If we were to employ, instead of the direct series, 



10} '^l^atbioas/ 1, Nj, Ng, Nj— 1, &c., 



the signaletically equivalent reverse series ^ ^ 



Ij, N., N„_,.N^~l,&c., ^^ 



the arithmetical difficulty would be much increased in consequence 

 of the quotients becoming rapidly more complex as the division 

 proceeds. It were much to be desired that some person practi- 

 cally conversant with the application of Sturm's method, such as 

 that excellent and experienced mathematician, my esteemed 

 friend Professor J. R. Young, would perpend and give his opi- 

 nion upon the relative practical advantages of the two methods of 

 substitution ; the one that where the residues are employed, the 

 other that where the quotients. 



I ani bound to state, that but for a valuable hint furnished to 

 me by my friend, that most profound mathematician, M. Hermite, 

 who discovered a theorem virtually involving the transformation 

 of Sturm's theorem here presented, but founded upon entirely 

 different and less general considerations, and in the origin of 

 which hint, as arising out of my own previous speculations upon 

 which I was in correspondence with M. Hermite, I may perhaps 

 myself claim a share, this theory would probably not have come 

 to light. It is of course not confined to Sturm's theorem, which 

 deals only with the special case of two functions, whereof one is 

 the first derivative of the other. 



There is a larger theory, to which M. Sturm's is a corollary, 

 which contemplates the relations of the roots of any two func- 



