Royal Irish Academy. 535 



This process, founded upon the dialytic principle, admits of a very 

 simple modification. Let us begin with the case where » = 0, or 

 m = n. Let the augmentatives of U be termed U , Up U 2 U 3 , .... 

 and of V, V , V„ V„, V 3 , .... the equation themselves being written 



\J = ax n + bx n - 1 + cx n ~ 2 + &c. 



V = a'x n + b'x n ~ l + c'x"- 2 + &c. 



It will readily be seen that 



a' . U — a . V , 



(i'U -*V ) + (a'U 1 -aV 1 ), 



(c'.U -c.V ) + (J'U l -6V I ) + («'U s -aV i ),&c. 



will be each linearly independent functions of x, x % , x m ~ l , 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 m ~~ l , x m ~ 2 , . . . . x m ~~ *+*. Thus, to obtain the final de- 

 rivee, 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 — i. The 

 equation V may be conceived to be of n instead of m dimensions, if 

 we write it under the form 



. x n + . x n ~ l + . **-* + + , x m + 1 



+ ax m + (3x™-l + 8ic. = 0. 



and we are able to apply the same method as above ; but as the first 

 / of the coefficients in the equation above written are zero, the first 

 i of the quantities 



(a' V -a V ), (b> U - b V ) + («' U, - a V,), &c. 

 may be read simply 



- a . V , -J.V -oV„ - c V - 6 V, - a V 2 , &c. 



and evidently their office can be supplied by the simple augmenta- 

 tives themselves, 



V = 0, V,=0, V 9 = 0.... ^ = 0; 



and thus < letters, which otherwise would be irrelevant, fall out of 

 the several derivees. 



The author then proceeds with remarks upon the general theory 

 of simple equations, and shows how by virtue of that theory his me- 

 thod contains a solution of the identity 



X r .U + Y r .V = D r ; 

 where D r is a derivee of the rth degree of U and V, and accordingly, 

 X r of the form 



X + px + vx- + + fla B »- r - 1 , 



and Y r of the form 



I + mx + .. .. + tx n ~ r - 1 , 

 and accounts a priori for the fact of not more than (» — r) simple 

 equations being required for the determination of the (m -f- n — 2 r) 

 quantities A, p, v, &c. /, m, n, &c, by exhibiting these latter as known 



