DERIVATIVES ASSOCIATED WITH LINEAR D1K1 IIIJKM I Ah EQUATIONS. 417 



class is taken to be the common degree of all the invariants of the class. In each 

 class the invariants are of two kinds, viz., composite, these invariants being expressible 

 in terms of invuriunt.s of earlier classes; and proper, these not being expressible in 

 such terms. The number of proper invariants in any class above the second is (n 2) 8 ; 

 but only n 2 of this number are quite independent of one another, and the remain- 

 ing (n 2) (n 3) proper invariants of the class can be expressed in terms of one 

 (or more) of the independent proper invariants and oT invariants of lower classes. 

 And the following are the proper invariants of the classes in succession : 



First, the priminvariants 6 8 , 6 4 , . . . , 6,, . . . , B, each of which is linear in the 

 coefficients of the differential equation, supposed reduced to its canonical form, and 

 their derivatives ; the index of each invariant is the same as its subscript number ; 



Second, (i) the quadriderivative functions 



. , = 2(7^8",- (2cr + l)e'J, 



which are n 2 in number (o- = 3, 4 ..... n) and are independent of one another ; 

 the index of e,,, is 2tr + 2 ; and (ii) the (n 2) (n 3) Jacobians 



of index X -j- p. -f 1 (X, p. = 3, 4, . . . , n), but only n 3 of these are independent, 

 and the remainder can be expressed in terms of these n 3, properly chosen, and of 

 priminvariants. The two kinds of proper invariants in this class are algebraically 

 independent of one another ; 



Third, there are n 2 independent cubinvariants given by 



6,. , = <**,&. , - (2<r + 2) e,, ,6', 



of index 30- + 3, and there are (n 2) (n 3) proper cubinvariants dependent on the 

 foregoing n 2 ; 



Fourth, there are n 2 independent quartinvariants given by 



e,, , = crf^e',, 2 - (3o- + 3) e,, .e', 



of index 4<r + 4, and there are (n 2) (n 3) proper but dependent quartinvariants ; 

 and, generally, the rth class contains n 2 independent proper invariants given by 



,,r- 2 - (r - 1) (a- + l) 0,, r _ 8 e', . . . , . (xi.) 



of index ? (a- + 1), and also (n 2) (n 3) proper but dependent invariants. 



And all the invariants of the 7-th class, for every value of ?, are of degree r in the 

 coefficients of the differential equation and their derivatives. 



50. In this connexion two points remain to be noticed. It has already ( 7) been 

 remarked that M. HALPHEN has, for the quartic, derived a series of invariants from 



MDCCCLXXXVIH. - A. 3 H 



