Mr. J. Cockle on the Method of Symmetric Products. 135 



59. The quantity between the first brackets is a pure epimetric 

 function, whose structure is 



a,{t)=:t„ e^[t) = t;\ eS)=='t:\ e^{t) = tT\ e^{t)=t;\ 



and for which (lb. p. 446, Art. XI.) 



eA/r(e) = e(e-l)=3x2 = 6, 



a result which confirms our previous detei'mination of the num- 

 ber of values of e{x). 



60. In the present state of the subject of equations of the fifth 

 degree, I do not desire to have any remark of mine placed in a 

 higher category than that of conjecture. I shall be satisfied if 

 they have a sufficient degree of probability to merit further inves- 

 tigation. 



61. It would, then, seem that the epimetric e{x) is susceptible 

 of finite algebi'aic evaluation ; and this for two I'easons. 



62. First. The six quantities s are not independent, but are 

 functions of the five quantities x. They do not, therefore, con- 

 stitute the roots of a general equation of the sixth degree, and, 

 being subject to the internal relation {t), theii- determination is 

 facilitated. (Vide Poinsot, loc. cit. sup.) 



63. Second. The product ir^{x^ is one of those 'critical' 

 functions which, under their symmetric form, I have defined in (5). 



64. We have, in fact, 



a{x) = c,G, + fi.Z-\dp^^- 7p,p.,+ \Qp,)% 



and, the unbracketed quantity being critical, [b) can only enter 

 into e{x) through the bracketed expression, which is rational. 

 Hence the mode in which b is involved in s is such as to free 

 the former quantity from radicality, and to indicate relations 

 favourable to solution, or rather, perhaps, which account for its 

 existence, supposing it to exist. 



65. Let us now proceed to an equation in y, connected with 



* Replacing y by a? in (5), it may be well, througbout the whole of this 

 discussion, to consider the coefficient of p,, in Cj; as unity, and, conse- 

 quently, c', Cj, C3, and C4 as respectively ccpial to 4, —.3, — 8 and — S^S"'- 

 We shall then have tlie result given in (()4), from which it appears that the 

 R of f8) is not eqiial to U4. 



That result may be readily verified, for in Mr. Jen'ard's last notation 

 (Phil. Mag. for May 1853, p. 355) the relation of (38) becomes 



P,=©4-(&1 .3+1 ((S22)4 (5 P . 2 . 3 - JL *&1^ ; 



and this, by the table at p. 33 of his ' Researches,' is equal to 



7?,^— 5;7,Vj+5;;/+.''j7j,/)3— 15/;4, 



and the substitution of/; for B in (41) will afford the means of conqdeting 

 tlie veriticatiun. 



