498 Rev. B. Bronwin on the Integration and 



(d z u du\ 



X \d^ + m d~ X ) +nU= °> 

 which is more simple than the given equation. 

 The equation 



d' 2 y /dy \ 



d^ +rx {div + m y) +r y=°> 



2 u , (du \ n 



reduces to 



d*u 



d. 

 where y— (D-\-m)u. 



With the same transformation we have 



,r 2 -f^ = (D-\-m)(Dx 2 u — 4fXu) + 2mxu-^2u 



x*y = (D + m) x 2 u — 2 x u ; 

 and we shall find 



to transform immediately into 



d 2 u (du . \ _. 



Or to take a more general example, 



( j^+( rx% + nx + k )\j^ + »*#)+ (2 r* + n),y=0 



will transform to 



^ + {r ^ + n, + k)(^ + m u)=0. 



In the two last examples, where y= (D + m) «, the trans- 

 formed are more simple than the given equations. 



To give one more example of this sort of transformation, 



(P 11 d y 



let -y-4 +■ mx 2 -y% — 2my=0. 



d x d d x* 



Here we shall make y—Xlu + mx* u. Then 



%£ =D 4 « + «D 3 ^«= D 4 m + mx*TPu + QmxWu + 6?nDu 

 dx 6 



= (D + mx q )D s u + 6mx D 2 u + 6 mD w=(D + m # 2 )D 3 u 

 4- 6 m D 2 x u—6 m D u. 

 Also 



#2 ^ = tf J)3 g + m x <2 D2 ^2 M = J)3^ w _ g Wan _|_ 6 D 8 



rf.r 2 

 + w# 2 D 2 # 2 m= (D + ?« x 2 ) T) 2 x-u-6 D*xu + 6 D k. 



