458 MR. A. R. FORSYTH ON INVARIANTS, COVARIANTS, AND QUOTIENT- 



Ci 8 11934 



1 '.: 



79 3.5~i~Ta865~9 



:; :t V ', 



T8 e 



, , 



~ 



+ i 

 TTS^T 



iu 



1333 



335 8' 4 



72 



Ta 



(42). 



These values show that all the priminvariants (and therefore all the derived 

 invariants) of the associate sextic are included in the invariants of the original 

 quartic ; and since the variable of the sextic is covariantive, and is included among 

 the covariants of the given equation, it follows that all the covariants, identical and 

 mixed, of the associate sextic are composed of covariants and invariants of the original 

 quartic. Hence, the theorems of 62 are verified for the linear quartic. 



108. There are many other equations possessing covariantive properties similar to 

 those in the associate variables ; among such equations are those, for instance, which 

 have their dependent variables composed of one or more than one of the aggregate 

 of dependent variables, original and associate. Thus the equation, which has 

 for its dependent variable the square of the dependent variable of the equation of 

 order n, is of order %n (n + 1), and all its invariants are invariants of the original 

 equation ; and the reduction of such an equation, when obtained, to its canonical form 

 will be very similar to the reduction to its canonical form of the associate equation 

 which has, for its dependent variable, the variable associate of the first rank of the 

 equation of order w -f 1. Thus, for instance, if we write t = u~ where 



= 



it is easy to prove that the equation in t is 



dz* v ' ' dz 



and the verification that the priminvariants (and therefore all the concomitants) of 

 this equation are included among the invariants of the quartic would proceed on lines 

 very similar to those of the verification for the quartic. 



109. For the general differential equation of order n, the equation satisfied by the 

 quotient of two solutions is of order 2n 1 ; a knowledge of n 1 special solutions 

 Xj, X 2 , . . . , X, ( _! gives the primitive in the form 



. . _ A + 

 " 



+ . . . + 



BO 

 and leads to the derivation of n particular solutions of the original differential equation 



in the form 



_j _i _i _i_ 



where 



t -l) x(-2) 



.-1) 



,-2) 



\ ( - i) \(.n- V 



A ,.A ...... 



n I ' 1 ' 



