Mr. J. J. Sylvester on Aronhold^s Invariants. 367 



additional resistance brought into its path. It will appear to be 

 an intense current according to the older mode of expression. 



On the other hand, when the same conducting body is intro- 

 duced into a circuit of small necessary resistance, the whole sum 

 of the resistances of the circuit may be greatly increased by the 

 addition of the fresh resistance to the original necessary resist- 

 ance ; and as a consequence, the current will be diminished in 

 the same proportion. The magnetic needle will return to the 

 zero-point, and the remaining current will produce no effect on 

 the conducting body. In short, the current arising from a 

 source of small necessary resistance will appear unable to over- 

 come the additional resistance brought into its path. We have 

 here a current of small intensity according to the older mode of 

 expression. 



If it be now granted that Ohm^s theory gives an explanation 

 of the phsenomena which heretofore have been explained by the 

 hypothesis of two different qualities in the electric current, then 

 as it is one of the first and most important principles of induc- 

 tive science not to assume two causes for any effect when one is 

 sufficient, and not to assume without the most urgent necessity 

 the existence of new qualities, it appears desirable to lay aside 

 the older hypothesis of intensity and quantity, and to adopt instead 

 Ohm^s theory of sources of electricity having different degrees of 

 necessary resistance. 



LX. A Proof that all the Invariants to a cubic Ternary Form 

 are Rational Functions of Aronhold's Invariants and of a connate 

 theorem for biquadratic Binary Forms. By J. J. Sylvester, 

 F.R.S. 



[Continued from p. 303.] 



NOW let us proceed to Aronhold's famous S and T, the in- 

 variants to the general cubic function (^, y, z)^, forms 

 equally dear to the analyst and geometer. (Vide Mr. Salmon's 

 Higher Plane Cnrves passim.) 



The method will be precisely the same as that applied to 5and/*. 

 We commence with the canonical form 



a^ -hy'"^ + s^ -{-Qm xyz. w? 



On substituting x-\-y-\-Js, w+py + p^z, x -\- p^y -\- pz for x, y, z, 

 where p is the cube root of unity, the above quantity takes the 

 form (3 ^ 6^) (^.3 j^ySj^^j^ 6^(^) . ^^yr), 



where ^. . _^ I86-- 18m_ J^--m 



• * The s is Mr. Cayley's property, the t belongs to Professor Boole, 

 having been by him imparted, in the infancy of the theory, to Mr. Cayley, 

 by whom it was first given to the world, at least in its character as an 

 Invariant. 



