184 Mr. J. Cackle on the Method of Symmetric Products. 



essentially from those of Lagrange. Abel {CEuvres, vol. ii. 

 pp, 185-209) never desisted from the efforts commenced by 

 Ruffini (lb. p. 186). The joint argument of Abel and Sir W. 

 R. Hamilton appears to have been assented to by Mm'phy (Equa- 

 tions, p. 77), but grave doubts respecting it have been thrown 

 out by Dr. Peacock. Professor J. R. Young* inclines to their 

 conclusion. In favour of the possibility of the solution we may 

 cite Ivory (Equations, Encycl. Brit. 7th ed. vol. ix. p. 341), 

 and Mr. G. B. Jei-rard. Mr. Bronwin wavers in opinion. Poinsot 

 regarded the question as involved in utter uncertainty (Preface 

 to the 3rd ed. of Lagrange's Equations, p. xvi.). Vandermonde, 

 whose theoi-y of equations Lagrange (Equations, p. 273) con- 

 sidered as being, to some extent, more direct than his own, after 

 some elaborate investigations, states that he is not in a condition 

 to offer even a conjecture upon the possibility of the general 

 solution (Par. Mem. for 1771, pp. 365-416; see Art. XXXIV. 

 of his paper) . 



54. In these papers we have been conducted d prioi'i to La- 

 grange's functions, as those which seem to fulfill the conditions 

 of maximum symmetry. Equations of the fifth degree furnish 

 us, not with a symmetric, but, with what, in the nomenclature of 

 my Fragment on Multiplicity of Values f, may be termed an 

 epimetric product. 



55. Thus, X being the I'oot of a general quintic, we have 



'7ri{x^ = a{x)-\-e[oc), 

 where a is symmetric and e epimetric. 



56. Let S], ^2, . , Sg be the six values of e{x). Then, this 

 function being hyposymmetric, it may (Phil. Mag. Dec. 1853, 

 p. 448, Art. XX.) be expressed in terms of A'g and of four quan- 

 tities /p /g, ^3, ^4 connected with x by the relations 



fjSro?^ *'5> tci^'=^Xi2 X^, t^ = X^-^X^j t^-=X^ X^', 



or may, in other words, be represented by 



e^{x) + (j){x^). 

 67. We have, consequently, six equations of the form (Ibid.) 



and, if by means of five of these (f> and t be eliminated from the 

 remaining one, we obtain 



X(si, S2>-,«e)=0 (l) 



58, Replacing a, /3, 7, B by 1, 2, 3, 4 respectively, we find 



* Equations, pp.468, 4(V.) ; Mechanics' Magazine, vol. xlviii. pp. 101, 

 102; On the General Principles of Analysis, pp. 4y-51. 

 t See Phil. Mag. for December 1853 (S. 4. vol. vi.), pp. 444-8. 



