Transformation of certain Differential Equations. 499 



Substituting these values, the proposed equation becomes 



{D + mx*) {D 3 u + mD 2 x*u-2mu) = 0; 



or D 3 M + mD 2 # 2 «— 2 mu=Q; 



d A u od 2 u du 



or -=— „ + m x l -r-T. + 4 m x -7— =0. 



Make -7- = 2, and the preceding becomes 



</# 



^2 a d z 



-7—0 + »* ar -7— + 4 ?» x z ss 0, 



or a .r 



which is only of the second order. The integration of it will 

 gives. Then y = (D + mx' 2 ) D -1 z. 



Positive powers only of D have been used in these trans- 

 formations ; but it is obvious that we may employ negative 

 ones also, and may make such assumptions as j/ = D~ l u + m u, 

 &c. We may also use higher powers of D, and we may ex- 

 press y by more terms than two or three. And it is plain that 

 we may apply a similar method to the transformation and in- 

 tegration of equations in partial differentials, and also to equa- 

 tions in finite differences. I shall close this paper with some 

 examples of a more simple transformation. 



Let ( 1 -^)^-i 3 (^- 1 )(^-2)^/=o. 



Make y = D p ~ 3 u. 



Then Dp u-x 3 Dp u—p(p—l)(p — 2)DP- 3 u=0 t 



which by (a.) reduces to 



Dp u-Dp X 3 u + 3 p Dp- 1 x*u— 3p{p — 1)Dp-*xu = 0; 

 or 



D 2 w— D*a?u+$pDx*u-3p (p-\) xu = D~p +2 = X 

 suppose, which is equivalent to 



(1 -^ 5T 2+( ^- 6) * 2 Jx " 3 C*- 1 ^- 2 ) xu = x ' 



which is only of the second order. 



Let (1 -x 3 ) d ^ + p(p+]) (p + 2)y=0. 



Make y — D~P- 3 u. 



Then D~p u-x 3 D~Pu+p{p + l) (p + 2) D~P' 3 u = 



by (b.) will reduce to 



D-pu— D'Px 3 u— 3pD-P- x x*u — 5p (p+l)D-^- 2 d:M=0; 



or D i u-D 2 x 3 u-3pDx s U"3p {p+l)xu = 0; 



