On Contrariants, a New Species of Invariants. 375 



and will be of the deg . degs n, ; %— >1, 1; n—2,2;... 

 1, (ft— 1); 0, n respectively. 



I shall show that the same two forms, when treated as con- 

 travariantive, are also n + 1 in number ; viz. two, as before, 

 of the deg . degs n, 0; 0, n; and the rest of the deg . degs 1,1; 

 2,2; 3,3;...; <>-l), (w-1). 



The proof is instantaneous, or nearly so ; for by two con- 

 tragredient substitutions, / and <f> may be transformed into 

 a x x\ + a 2 x\ + . . . + a n x\ and k%\ + k%\ + . . . + kp n (where the k is 



employed in lieu of unity as a safeguard for maintaining 

 homogeneity in the subsequent operations). If, now, 



d d d 



dwi dx 2 ' ' ' dx n 

 be written in lieu of 



f 1? ?2j • • • in 



in the second form, and this form so modified and its succes- 

 sive powers be made to operate upon the first form and its 

 corresponding powers, it is obvious that by combining the 

 results of these operations, we may obtain the invariants 



2&a x ; SAVi^ ; • • • ; %k n ~ l a 1 .a 2 .. . « re _i; 

 in addition to which we have the invariants of the two forms 

 taken separately, viz. a x . a 2 . . . a n and k n . 



These n + 1 invariants are, upon the face of them, mutually 

 independent. 



But any invariant of the two forms must be symmetrical in 

 regard to the a's, and consequently must be a symmetrical 

 function of 



£fo&ij XJc 2 a 1 a 2 ; . . . ; Xk n ~ 1 a 1 . a 2 . . . « w -i; «i • a 2 . . . a n , 



say F; or at all events of the form k ±e F, where 6 is an integer 

 less than n ; but in such case k ±e will be a rational (I do not 

 say integral) invariant of the system, and therefore of the 

 second form of the system, which we know has no other invariant 

 than powers oik n . Hence 6= 1, and consequently there are no 

 invariants other than the n + 1 independent ones above given. 

 These, then, are the irreducible contrariants ; and it may be 



noticed that the sum of their deg . degs is — - — > — ~ — , the 



same, i. e., as the like sum for the irreducible invariants of 

 the same two forms regarded as covariantive. 



An ordinary invariant might, in analogy with botanical 

 language, be termed Invarians Vulgaris, or Simplex, or 

 Eisensteinensis (from the name of the first discoverer of the 

 species) ; and so a Contravariant, which necessarily belongs 



