DERIVATIVES ASSOCIATED WITH LINEAR DI I I KIIENTIAL EQUATIONS. 391 



Similarly 



C,, ,_ 8 = A ( ~ 3) '- (22^- + 4 (n - 4) *W + (n - 4) (n - 5) *"} 



= A ( ~ 3) *'"~ 8 (2Z* + (4n - 10) ZZ' + (n - 5n + 6) Z 8 } ; 

 C._ u ._, = A (n - 3) z'- 5 {4zV + 3 (n - 4) 2 " 3 } 

 = a L 4 (" ~ 3) 2'- 8 {4Z' + (3u - 8) Z-} ; 

 C._ 8 , _, = $ (n - 3) '-* 2" = fc (n - 3) z'- 8 Z. 



By means of these (7)' changes to 



Xz-'Qa = ^ (n - 3) X {2Z" + (4n - 10) ZZ' + (n* - 5n + 6) Z 8 } 



+ A ( - 3 ) { 4Z' + ( 3 n - 8 ) Z 2 } X' + . ( n - 3 ) Z (\" + P a \) 



reducing on substitution for X'", X", X' to 



Equations (6) and (7) agree with the equations given by BRIOSCHI for the case of 

 the cubic (n = 3) and that of the quartic (n = 4) ; the elimination of Z between them 

 is in process precisely similar to that in those special cases, and it leads to the result 



14. It appears from this investigation and from the results of BRIOSCHI and 

 HALPHEN that there are rational integral functions of the coefficients of the diffe- 

 rential equation and their derivatives such that, when the same function is formed 

 for the transformed differential equation, the two functions are equal save as to an 

 integral positive power of z. These functions are called invariants ; the exponent of 

 the power of 2' may be allied the index of the invariant 



Dimension- Number ; Homogeneity. 



15. The index of an invariant can easily be settled by the following considerations. 

 We can assign to each coefficient of the differential equation a certain number, called 

 for this purpose its dimension-number, suggested by the similarity with the theory of 

 dimensions of homogeneous functions.* For the present the dependent variable y will 



* This process is practically identical with M. HALFBKN'S assignation of weight ; see above, Historical 

 Introduction, 6. 



