Mr. J. Cockle's Analysis of the Theory of Equations, 509 



far as its own processes are concerned, but, in the form in 

 which it is usually exhibited, an assumption is made that we 

 are acquainted with the roots of a cubic, and with the manner in 

 which those roots enter into the coefficients. I have some 

 time since (see Phil. Mag. S. 3. vol. xxxii. pp. 421, 422) 

 given one or two corrections and additions to my former letter; 

 but I must here remark that Vandermonde should share with 

 Lagrange the glory of the discovery of a general analysis of 

 equations, and that to Lagrange is due a mode of representing 

 the roots of equations, which some have attributed to the late 

 distinguished Murphy. Connected with, and presenting some 

 general resemblance to, the methods of Lagrange and Vander- 

 monde, is my method of symmetric products respecting which 

 you will find full information in my " Notes on the Theory of 

 Algebraic Equations." The connexion and resemblance con- 

 sists in this; that the linear functions whose product, in my 

 method, is symmetric, are the same as those whose squares, 

 cubes, and higher powers have, in the methods of Lagrange 

 and Vandermonde, only one, two, &c. values respectively. I 

 should wish those " Notes " to be read in connexion with this 

 letter. 



I remain, my dear Sir, 



Yours most truly, 

 To T. S. Dames, Esq., F.R.S.L. $ E. James Cockle. 



tyc. Sfc. fyc, Woolwich. 



Postscript. — October 25, 1850. Let me add — 

 (33.) That, besides certain problems alluded to above, we 

 may have that of the cubic solution of three tertiary quadratics, 

 a solution which might become useful sometimes. That I 

 have given a curious example of the depression of a certain 

 form of equations, at pp. 45, 46 of the recently published 

 Supplementary number of the Mathematician ; and that, with 

 respect to the solution of equations of the fifth degree, I have 

 at present nothing to add to the remarks which I made in the 

 Philosophical Magazine for December 1849 (S. 3. vol. xxxv. 

 pp. 436, 437). Similar obstacles to that presented by equa- 

 tions of the fifth degree occur in other departments of know- 

 ledge. In the theory of elliptic functions, for instance, inves- 

 tigation was arrested by the impossibility of making a sum of 

 two integrals fulfill certain conditions. But Abel obviated this 

 by employing a sum of more than two integrals. I do not 

 mean to say that the equation of the fifth degree is solvible, 

 but I must protest against its supposed insolvibility checking 

 all inquiry in that direction. Those equations are probably 

 destined yet to fill an important place in science. 



