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A Class of Functional Invariants. 



[Mar. 15, 



^=,| + 22 | + 2 g + 2 . S | + 3,|) + .... =0, 



= +.<g r + 3^ f + eg- -t- . . . = 0, 



a i 30 , Q 80 , 30 30 30 .30 n 



and two index-equations, viz. : — 



3X0 = 2^ + ^ + 4^ + 3^ + 2^ + .... 



Op oq Or os Ot 



3X,= ,| + 2 |+^+4 + ^ + ...., 



in the last two of which A, is the index, an integer determinable from 

 the form of by inspection. In these equations p and q are the partial 

 differential coefficients of the first order ; r, s, t those of the second 

 order ; a, b, c, d those of the third order; and so on. 



An invariant is said to be proper to the rank n, when the highest 

 differential coefficient of z occurring in it is of order n. By means 

 of the solutions of the equations A x = A 2 = A 3 = A 4 = 0, con- 

 sidered as simultaneous partial equations, and by using the remaining 

 equations, the following propositions relating to irreducible invariants 

 in a single dependent variable z are established : — 



Invariants can be ranged in sets, each set being proper to a parti- 

 cular rank ; 



There is no invariant proper to the rank 1, and there is one, viz., 



(fr — 2pqs+pH, proper to the rank 2 ; 

 There are three invariants proper to the rank 3 ; 

 For every value of n greater than 3, there are n + l invariants 



proper to the rank n, which can be chosen so as to be linear in 



the partial differential coefficients of order n. 



Every invariant can be expressed in terms of this aggregate of irre- 

 ducible invariants ; and the expression involves invariants of rank no 

 higher than the order of the highest differential coefficient which 

 occurs in that invariant. 



A special class of invariants, proper to ranks in numerical succession, 

 is given by combinations of A , A l5 A 2 , .... where — 



