133 Mr. J. Cockle on the Method of Symmetric Products. 



vit, in terms of (48), deduce 



E2_E-2=0, 



aud find that — 1 and 2 are the values of E. 



50. The substitution of 2 for E leads to nugatory results, but 

 that of —1 changes {k) into 



^2 + .. + 1=0, 



the roots of which are the unreal cube roots of 1. If we repre- 

 sent these quantities by « and «^, the condition 



H-S.«,^2=0 

 points to the relations 



u^=^^=u and «2=/8j=a^. 



51. Denoting the transformed equation in y by 



we shall find 



Y,Y^=ir^{xs-^)-' = q-^{a^-3b)^^+{ab-9c)^+ {b^-3ac)}, 



and, if we determine ^ so as to make 'TTc^iyQ) vanish, the relations 



Yi(orY2)=0, t.y = q^, ^.y{y^ = q^ 

 will give y and, consequently, x^. 



52. The Method of Symmetric Products enables us to confine 

 the attention to a greatly diminished number of functional 



* The reducing equation of (.51 ) was fii'st obtained by Bezout (Pai*. Mem. 

 for 1/62, p. 24). I subsequently (Camb. Math. Joum. for May 1841) was 

 conducted to it by radicallj^ different considerations. Mr. Cayley (Camb. 

 and Dub. Math. Journ. for May 1851) has, by giving a coefficient to a^, 

 generalized my process, and Mr. Rotherham (Eng. Journ. of Ed. for August 

 1853) has given a solution of a cubic, independently arrived at, but substan- 

 tially identical with mine. The reducing equation is that which arises 

 from making T^^iyi) vanish. 



The intrinsic interest of Mr. Cayley's result is enhanced by its connexion 

 with Dr. Boole's functions 6 and &, and with his own researches on hyper- 

 determinants. The generahzed form of tt^ is the function employed by 

 M. G. Eisenstein {Crelle, for 1844, vol. xxvii.), and adverted to by M. C. 

 Hermite (lb. for 1851, vol. xli.) as a " quadi-atic form" throwing light on 

 the structure of cubic equations. 



It is strange that the function ttj should have been so much unnoticed. 

 Its coefficients were present, and that too in a suggestive form, to Lagrange 

 (Equations, p. 42) as they had been to La Fontaine (lb. p. 141). Compare 

 Waring {Misc. Anal. p. 22, line 7), Gamier (p. 115), Rutherford (Com- 

 plete Solution, &c. p. y), and Mech. Mag. lii. 229, where for M read — M. 



Various direct solutions, including those of Tartaglia, Ivorj', Graves, 

 Waring and Rutherford, and two of my own, are adverted to in my ' Notes.' 

 That of Waring {Med. Ale/, p. 98) has been erroneously attributed to Laplace. 

 Some assumptions suggested, though not for the same pui-pose, by Newton 

 (see Fluxions, Lond. \/37>\>- 23) may be compared with that of Bezout. 

 Dr. Rutherford's solution, characterized by his usual skill, has a general 

 resemblance to that of Bezout. 



