131 



x\U = 0, xKUzzO, x-.U=0,....x'-\U=0, 

 x\F=0, xWzzO, x^.V-Q, a;'+*-'. F=0, 



we may eliminate linearly 2s -\- l — 1 quantities. 



Now these equations contain no power of x higher than 

 m ^ I -\- s—\ ; accordingly, all powers of x, superior to 

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

 {m — 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 V, 

 and eliminate [m + n—X) quantities, namely, 



X, x^, x^, up to «'"+"-'. 



This process, founded upon the dialytic principle, admits 

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

 where i — Q, or m — n. Let the augmentatives of U, be 



termed U^, Uy, U^, U^, and of F, Vo, F„ F^, F^, 



the equations themselves being written 



U = aa;« + bx"-^ + cx"-^ + &c. 

 F zz aV + b'x"-^ + c'x"-^ + &c. 



It will readily be seen that 



a'.Uo-a.Fo, 



(b'Uo-bFo) + (a'U,~aF,), 



(c'.f/o-c.Fo) + {b'Ui -bV,) + (a'a^-aV,), 



&c. 



will be each Knearly independent functions of x, x'^, 



a;™-', no higher power of x 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'"'—\ x'"—^, .... 

 3jm-s+i Xhus, to obtain the final derivee, we must make use 

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



Now, let us suppose that i is not zero, but m — n ~ u 



VOL. II. M 



