468 report — 1878. 



the final term being c„y „ or c r y », according as m or r is the greater. In the 

 former case (9), and in the latter (10), is the more convenient form. 



The substitution ( J/**' ) changes the algebraical equation (1) into 



\—r, m r, (j 



ay m + bxy r +c = o, (11) 



and the same substitution changes the differential equation (9) into 



— r m m—r 



[r d n~\ T d ~] , . m ~ r b m c r T~m-r d n , ~l 

 -L x "~ 7 ±\ a?— [«„=(-) x- 1 x u n , 



m dx mJ L dx J a r c m L m dx m J 



which, since 

 may be written 



m—r 



L dxJ a r c m \-m dx m J L m dx m J v ' 



This confirms a result obtained by the author some years ago by the aid of La- 

 grange's theorem, viz., that the differential equation 



n n—r r 



X d~\ Vn—r d ,m -.~\ T~ r d m -. ~f „ _ 



a \ x— I m = I x — + — — II \ - x _ — 1 I x n r u 



L dxJ L n dx n J L.n dx n J 



is satisfied by the w th power of any root of the equation. 



yn —xy n ~ r '+a = o. 



See Boole's ' Differential Equations,' Supplementary Volume, page 199, where the 

 reader should correct a misprint in the seventh line from the top of the page ; the 

 factor x u should be x n ~ r . 



When m and r are not prime to each other, a reduction in the order of the 

 differential equation may be effected. For, if m = ap, and r = ap, a being the 

 greatest common measure of m and r, the condition pm = qr is satisfied by making 

 p-p and q = p, and we have, corresponding to the equations (9), (10), and (12) the 

 three following, viz : — 



L/x — p dx a(p — p)-l L dx-1 



-(-yff-f jl. < • r i]V%, ( i3) 



b»C lp — p dx a{jx — p) ~i 



r m x d «_"|Yir-Y 



Lp—p dx a(p -p)-l L dx-1 



p6^ r^_ x a _ «_ _ nV-^, (U) 



a'c* Lp — /*. dx a(p—p) -I 



either of which is satisfied by the n th power of any root of the algebraic equa- 

 tion (1). And 



I ' x A-f Un =(-f±*rp.x* + « -ifrezp*'-»-i"TV« b d5) 



L dx_\ aPc^Lp. dx ap J L p. dx ap -J 



which is satisfied by the w th power of any root of the algebraic equation (11). 



The "differential resolvents" of (1) and (11) are formed by making w = l, in 

 which case each of the differential equations reduces to an order lower by unity 

 than itself, in accordance with the theory of differential resolvents. The author 

 noticed other cases in which, for particular values of n, the differential equations 

 admit of reduction. 



The passage from the symbolical to the ordinary form of linear differential 

 equations was shown to involve no practical difficulty. 



p-fi 

 u„ 



