430 MR. A. R. FORSYTE ON INVARIANTS, COVARIANTS, AND QUOTIENT- 



66. The number of terms in this succession of identical covariants is given by values 

 of?' from 2 to n 1, so that the succession includes, besides u, the n 2 terms U 2> 

 U 3 , . . ., !!_!. When the value n is assigned to r, the resulting covariant is one 

 which involves d'u/dz", and which can, therefore, by means of the differential equation 

 satisfied by u, be transformed so as no longer to involve this differential coefficient. 

 It will then involve derivatives of u of order less than n, and also the seminvariant 

 coefficients Q ; such a covariant will be called a mixed covariant, because its expression 

 depends partly on the variable and partly on the coefficients. Further, every succes- 

 sive covariant derived by the Jacobian process can be similarly transformed, and will 

 then become a mixed covariant. There will be some limitation on the number of 



ndependent covariants of the mixed type thus obtained ; for the elements, so far as 

 concerns the dependent variable, are only n in number, being u, u', . . ., u <Jt ~^\ and 

 elimination of these quantities among more than n mixed covariants will lead to 

 relations involving covariants and functions of the coefficients of the differential 

 equation only. From the fact that the quantities, which occur in the result of the 

 elimination, and are not functions of the coefficients, are covariantive, it is a priori 

 probable that such functions of the coefficients as enter are invariants of the differen- 

 tial equation, or combine with the variables to constitute mixed covariants. 

 For example, in the case of the cubic equation, which is 



-^ -f & s u = 

 in its canonical form, and has @ 3 for its priminvariant, it is easy to show in general 



U 3 = (n - I) 2 wV" -3( n -l)(n-3) uu'u" + 2 (n - 2) (n - 3) w' 3 , 



so that, when n = 3, 



U 3 = (3 l) a u z u" = 4 w 8 @ 3 ; 



and there are, therefore, no proper identical covariants involving u alone, except u and 

 U 2 for the cubic. The corresponding investigations, which must be deferred until the 

 mixed covariants are obtained, are given in 76 for the quartic, in 77 for the quintic ; 

 and the general investigation for an equation of any order is indicated in 138. 



67. In the case when there is given, not a differential equation, but a differential 

 quantic of the form 



l n - r\ 



and we are seeking the identical covariants for the transformation which changes this 

 quantic to 



n\ fr-u 



* 



\ - 



r n 



