ORDINARY LINEAR DIFFERENTIAL EQUATIONS 



271 



Designating the transformed point of {x,y) by [x^, y,) wehave^'^ 

 the relations: 



p- 



(') 



ip=%P'^7G) 



by which (1) is transformed into another linear differential equa- 

 tion of the same order 



<^i {^'x^Vx^Px) =0. (8) 



By this transformation the equation 



J^ +i> x^=0 becomes 



0, 



or 



8(f> 



Sy 



s^ ■ IF ^ ^~Hx ' 'W 



S<^, 8x,_^8^ By, , S0, 5^, 

 8x 



(Hx 



\ ^^x 



Vx 



p 



(Hx 



\ ^^x 



8y 

 But, according to (7) 



+ 



8<^, Bx, ^ B<f>^ By, ^ B_^ B^ 



Bp, ' By 



Bp, 



X ^Px \ 



/ Bx I 



) 



+ 



= 0. 



Vx 



p 



Bp, 



{y-xp)'^ ^ Bx {y-^pY^ ^^ 

 ' ^yx _ 1 ^^x _ 



by 



P 



y' ^y {y-^pY' ^y {y-^pY 



Substituting these values in (9), we get from (9) 

 B<j> B4>,( 1 x\ 



(9) 



(1) S. Lie. Beruhrunerstransfonnationen, Vol. I, p. 23. 



