131 



x\U = 0, x\U = 0, xKU=0,....x—\U=0, 

 x».V = 0, xKVzzO, x'^.V = 0,....x'+'-\V=0, 



we may eliminate linearly 2* + * — 1 quantities. 



Now these equations contain no power of x higher than 

 fw -f t + s — 1 ; accordingly, all powers of x, superior to 

 m — s, may be eliminated, and the derivee of the degree 

 (jn — s) obtained in its prime form. 



Thus to obtain the final derivee (which is the derivee of 

 the degree zero), we take m augmentatives of U with n of F, 

 and ehminate {m + n— 1) quantities, namely, 



X, x"^, x^, up to x"'+"-'. 



This process, founded upon the dialytic principle, admits 

 of a very simple modification. Let us begin with the case 

 where i =z 0, or wj — w. Let the augmentatives of U, be 



termed U„ U,, U.„ U„ and of T, V^, V„ V^, F,, . . . . 



the equations themselves being written 



U = ax" + ia;"-' + cx"-^ + &c. 

 V = a'x" + b'x"-^ + cx"-"^ + &c. 



It will readily be seen that 



a'.Uo—a.Fo, 



(b'Uo-bFo) + {a'U,-ar,), 



[&M,-c.F,) + {b'U, -bV,) + {a'U,-aV,), 



&c. 



will be each linearly independent functions of r, x"^, 



a:"'"', no /«^//er power of a; remaining. Whence it follows, 

 that to obtain a derivee of the degree {m — s) in its prime 

 form, we have only to employ the s of those which occur 

 first in order, and amongst them eliminate x^~\ .r"""^, .... 

 ^m-ii+i Xhus, to obtain the^«a/ derivee, we must make use 

 of «, that is, the entire number of them. 



Now, let lis suppose that i is not zero, but m = n ~ i. 



VOL. II. M 



