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



12. The conditions requisite for the symmetry of the terms 

 in y'^ are 



1 = a^a^a^a^ = ^^^^^^^^ = '^I'ii'iz'ii = ^\KhK 



13. The conditions derived from the terms in y^^yr andj/ij^,^ are 



E=2;.a=2./8=2. 7=2.8= "1 



S . «,a2«3=2 . /3i^ci^3='Z . 7i7273=2 • ^i^A^ 



14. Those derived from the terms in yi^y^^ are 



F=2 . «i«2=2 . A/S2=S • 7iT2=2 • SA- 



15. Hence, each of the four expressions 



(«U «2) «3> «4)' Ou /Sjy ySg, ^4), (7u 72. 73J 74)^ (81,82,83,84) 



involves the four roots of the equation 



^-E23 + r^2_E.+ i = o (c) 



in some (hypothctically) determinable but as yet undetermined 

 order. That order must of course be excluded which renders 

 the values of Y equal, for in such case we should be led to the 

 relation 



7^4(2/5) = ?A 

 a nugatory result. 



16. The equation (c) is recurring, and its roots are of the 

 forms \, \~\ fJ', fJ'~^- We are, consequently, at liberty to start 

 with the assumptions 



«j = X, Uci=\-\ a3 = fJ', et^^fJ''^. 

 And here for the present at least I leave the discussion, with the 

 remark that if no evanescent form of U4 be discoverable, that func- 

 tion may possibly be found to possess the properties oid>. modulus 

 of the given equation. A theoi-y of conjugate equations may hence 

 arise. If Abel's argument be undisputed, it is hard to conceive 

 that the theory of equations should not admit of an extension 

 analogous to that which he himself gave to the theory of elliptic 

 integrals. If its validity be denied, we may pursue our present 

 course with more sanguine anticipation. The consequences of 

 supposing that U„ is equal to zero will be a subject for after 

 inquiiy. 



2 Pump Court, Temple, 

 November 1, 1852. 



Postscript. I shall perhaps be forgiven for adding, that in the 

 Philosophical Magazine for June 1843 (S. 3. vol. xxii. pp. 502, 

 503), I gave a solution of an imperfect cubic which is free from 

 at least one defect under which that of Cardan labours, — the 

 arbitrary character of the operation by which the indeterminate 

 result of substitution is broken up into two separate equations. 

 I have also there obtained roots in an unobjectionable form. The 

 process is one of great simplicity. 



