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XIV. On Quaternary Cubics. By the Rev. Geoege Salmon, M.R.I.A. 
Communicated by A. Cayley, Esq. 
Eeceived June 14, — Bead June 21, 1860. 
In the following memoir I propose to make an attempt at an enumeration of the 
invariants, covariants, and contravariants of a quaternary cubic, that is to say, of a 
homogeneous function of the third order in four variables, which geometrically repre- 
sents a surface of the third order. This memoir, then, will be in continuation of Mr. 
Cayley’s memoirs on Quantics, wherein a similar analysis is very completely performed 
for binary quantics as far as the fifth order, and for ternary quantics as far as the third 
order. 
In consequence of the great length of the formulae when the general equation is used, 
I work with Mr. Sylvestee’s canonical form 
where the variables are connected by the relation 
It will be found that the discussion of this form leads to some results resembling those 
obtained for binary quantics of the fifth order ; the one quantic being canonically 
expressed as the sum of five third powers, the other as the sum of three fifth powers. 
Mr. Sylvestee has calculated the Hessian of a cubic expressed in the form given 
above, but in order to obtain with facility new covariants, it is desirable that we should 
also be in possession of a contravariant. We should then be able to substitute differ- 
ential symbols for the contragredient variables, and should thus have an operating 
symbol by the help of which we could derive new covariants from those known already. 
Let, then, a, /3, 7 , £ be contragredient variables ; let us suppose that when the 
original function U is expressed in terms of four independent variables, we have got 
any contravariant in a, (3, y, ^ ; and let us consider what this will become when the 
function U is expressed in terms of five variables connected by a linear relation. Now 
the contravariant in question expresses geometrically the condition that the plane 
ax-\-^y-\-yz-{-^uA-^v should possess a certain connexion with the surface represented 
by U. This plane, expressed in terms of four variables, is 
(a— £)^r+(/3 — £) 3 /+(y— £) 2 :+(S— £)^j; 
and it is apparent that the contravariant in terms of five letters is derived from that 
expressed in terms of four letters, by substituting a, — £, (3 — £, y — £, ^ — £ respectively for 
