428 MB. A. E. FORSYTH ON INVARIANTS, COVARTANTS, AND QUOTIENT- 



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p II n p\ 



in each of the quantities y lt y 2 , . . . , y M> and of an additional unit dimension in each of 

 the quantities y p +i, y f + $, , y- Since the functional determinant of y lt y 2 ..... y, 

 is a constant, the part of the factor dependent on the [n 1 ! -f- p 1 ! n pi} 1 

 dimensions in the quantities y lt y. 2 , . . . , y n may be expected to be a constant ; the 

 later part may be expected to be a functional determinant of y p+l , y p + 2 > > U* 

 This last is a special value of the dependent variable of complementary rank, and 

 is the conjugate of the dependent variable t p _ l omitted in the construction of the 

 integrating factor. Hence it may be expected that the variable of the differential 

 equation, adjoint to that in t f , shall have as its variable t n _ p . 



I do not propose to attempt to give here, however, a rigid investigation of the 

 inferences just suggested. 



The equation of lowest order for which an adjoint exists is the cubic ; after the 

 formation of this adjoint equation, which will be effected later ( 82) in connexion 

 with the investigation of some questions about the cubic, the identity of the 

 covariants of the two equations will be evident. Similarly for the case of the quartic 

 ( 102-107). 



SECTION V. 

 IDENTICAL AND MIXED COVARIANTS. 



63. In the last section a set of n 1 dependent variables y, t 2 , t& . . . , t a-l has 

 been obtained which are algebraically independent of one another, and each of which 

 possesses the same invariant! ve property as the fundamental .invariants ; and, just as 

 was the case with the invariants, we can, by using the methods employed in 

 Section III., deduce other covariants from each of these dependent variables alone, 

 from combinations of them with one another, and from combinations of them with the 

 invariants. As it is desired to retain only those functions which are not composite, a 

 selection must be made as before. The forms of the functions will be destitute of one 

 of the characteristics of the invariants ; their indices depend on the order of the diffe- 

 rential equation, and the number expressing this order enters into the numerical 

 coefficients, so that these new covariantive functions vary from one equation to 

 another. 



Identical Covariants in the Original Variable. 



64. In this class are included all those functions possessing the invariantive 

 property, and involving the dependent variables alone or their derivatives, but not 

 the coefficients of the differential equation, when taken in its canonical form ; on 

 which account they may be called identical, or absolute. Beginning with the original 

 dependent variable, we have 



