DKKIVATIVKS ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 401 



M can be expressed in the form MB s e 4 , that is, a composite invariant of index 7. As 

 this is not a new function, it is omitted, as in the case of 8 8 ; and the quantity which 

 involves the other constant is then 



3*44f dP,/ rfl>,\ ,rfP,rfP,l, p2 1155n* + 6048* + 6909 , 



H "^r 5 iT^r ~aa(+ijF 



Invariants such as these which have one part linear in the coefficients of the 

 differential equation will, for brevity, be called linear invariants. 



General Form of Linear Invariants ; Canonical Form of Equation. 



26. The last few results suggest a general deduction, which can be derived directly 

 from the equations (iii.), as to the general form of linear invariants. From those 

 equations we have 



\-'Q W W 



*~ vfr _ " /-i i YY '-\ p i 



- . ^H n ni ~T~ _ 11 '-' +!> * < "T" 



so that, 





X (z''Q. - P 4 ) = sP^\' + | (?i - s) '- P,.^ + terms involving P,_j, P^ 3 , . . . ; 



whence, by (8), 



*"Q. - P. = *P~ i ( i ( n - s) Z - i ( n - 1 ) Z } + . . . 



= ^s (s 1) P,_! Z + terms involving P._ 2 , P,_s- 

 Tims, 



z'^Q,-! P^! function involving P^y P^_ 3 , .... 

 and, therefore, 



_ =1 + w'^iQ^jZ = function involving P^.., P._ 8 , . . . 

 ax 



Combining these two, we have 

 >> {Q- ~ i ( - 1) d ^: 1 } - {P. - i (* - l ) d -^r} = ^notion involving P._ 2 , 



z 



MDOCCLXXXMII. A. 3 F 



