406 Mr. J. J. Sylvester on a remarkable Discovery in tlie 



may be any hyperdeterminant, or (as I shall in future call such 

 functions) co- variant of this product, understanding P(a7, y) to 

 be a CO- variant of («r, y) when V{lx-\-my,nx-\-pij) stands in pre- 

 cisely the same relation to f{lx -f my, rue -^py) as P(a?, y) iof{x, y), 

 provided only that Ip—mn^l. For the relation and distinction 

 between co-variants and contra-variants, see a short article of 

 mine in the forthcoming Number of the Cambridge and DubUn 

 Mathematical Journal for this month. In endeavouring to apply 

 the method of the text to the Sextic Function 



aa^ + Gbx^y -f- I6ca/^y^ -f 2da;^y^ -f 1 5exY + Qccy^ + qf, 

 thrown under the form 



S(^a7 + ^y)H20eU^ 



where 



U = (p^x + q^y) [p^ + q^) {p^ + q^y) ^s^-\-s^ochj-^Sc^,xy'^-\-s^, 



I obtain the following equations : 

 asg—bs^ -f csi — <;feo= ^(1^625q%— 54^Q.9i52 + 12^1^) 

 bs^—cs^ + ifej —^^0= €(54^0^1 •% + Qs^^.Sq—SGsqSq^) 



CSg— dSQ + eS^ —fio = € ( — 5450^2 • ■^3 ~" ^^l • *2^ + 3^% • "^1^) 

 ^3 — ^^2 +/*l — ^^0 = ^( "~ 1 6250%^ + 54^05^ + 1 2^3^25) . 



In these equations, if we call the quantities multiplied bv e, L, 

 M, N, P, we shall find 



53L- ^s^M- ^s^N-\-Sq.?=0, 

 and 



where I denotes the determinant, or, as I shall in future call 

 such function (in order to avoid the obscurity and confusion 

 arising from employing the same word in two difibrent senses), 

 the Discriminant*, which is the biquadratic (and of course sole) 

 invariant of the cubic function 



SoX^ + SiX^y + Scixy'^ + s^. 



* " Discriminant," because it affords the discrimen or test for ascertain- 

 ing whether or not equal factors enter into a function of two variables, or 

 more generally of the existence or othenvise of multiple points in the locus 

 represented or characterized by any algebraical function, the most obvious 

 and first observed species of singularity in such function or locus. Progress 

 in these researches is impossible without the aid of clear expression ; and 

 the first condition of a good nomenclature is that different things shall be 

 called by different names. The innovations in mathematical language here 

 and elsewhere (not without high sanction) introduced by the author, have 

 been never adopted except under actual experience of the embarrassment 

 arising from the want of them, and will require no vindication to those who 

 have reached that point where the necessity of some such additions becomes 

 felt. 



