DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 405 

 (iii.) In d'Qi/dz', where s > 2 < <r 1 and r +* = <r, the terms involving /*" are 



(iv.) In rf'~ ! Q2/j2*~ 8 the term involving ft" is 



When these quantities are substituted and the coefficients of corresponding terms 

 are compared, the satisfaction of the equation requires the following relations : 



iB o-(o--l) = B^cr-l) 



iB 1 (cr-l)(r-2) = iB 3 {2(2o--3)} 

 i B 2 (o- - 2) (o- - 3) = i B 3 {3(2o- - 4)} 



B,_ 3 3 . 2 = i B,_ 2 {(<r - 2) (o- + 1)}. 



The last equation determines B,_ 2 , the coefficient of rf*~ 2 Qo/Jz*" 2 , which is not 

 required for our present purpose, because this term, though linear, will not occur in 

 the canonical form. The other equations give 



B! = 0- B , 



2 (2<r - 3) 





--- 

 '"' 2 . 3 (2<r - 3) (2<r - 4) *' 



4 ~~ 



2 . 3. 4 (20 - 3) (2<r - 4) (2<r - 5) 1J 



*~ 8 ~" 



2 . 3 . 4 . . . (<r - 3) (2<r - 3) (2r - 4) . . . (<r + 2) l- 



Hence, taking B to be unity and passing to the canonical form, we obtain the non- 

 composite linear invariant of index <r in the form 



r.-V. . . . w, 



