Mr. J. Cockle's Analysis of the Theory of Equations. 505 



which I made clear in a second and somewhat more detailed 

 explanation (Ibid. vol. xxxii. pp. 51, 52, Art. II.). And I 

 have given some developments on the subject of the pp. 132, 

 133 (just cited) at pp. 32-34? of the Supplement to vol. iii. of 

 the Mathematician. 



23. Although invalid for effecting the transformation of 

 biquadratics to a binomial form, the investigations, referred to 

 in the last paragraph, enable us to transform the general equa- 

 tion of the fifth degree to a trinomial form. [See Phil. Mag. 

 S. 3. vol. xxviii. pp. 1 32, 133, 395, with the details and improve- 

 ments comprised in article I. to II I. of pp. 50-52 of vol. xxxii.] 

 And my trinomial transformation of the equation of the fifth 

 degree appears to possess two advantages over that of Mr. 

 Jerrard, whose genius first discovered the transformation in 

 question; — (1) we only employ three unknowns, and, (2), the 

 symmetric functions of the roots of the original equation, which 

 in Mr. Jerrard's process (Researches, p. 77 el seq.) rise as 

 high as the thirtieth degree inclusive, in mine are only in- 

 volved as far as the twentieth. 



24. In the problem which we have just been considering, 

 we see that the solution of the equation of the lower degree 

 and higher order precedes, and renders possible, the trans- 

 formation of the higher degree and lower order, affording a 

 confirmation of the accuracy of our system of classification. 

 And this indeed would be, in all likelihood, much further 

 confirmed had I space to do full justice to all the various 

 questions discussed or suggested by Mr. Jerrard and Sir W. 

 R. Hamilton. The necessity, however, of observing some 

 limits prevents me at present from doing much more than 

 refer to my own labours, with a deep appreciation of, and 

 interest in, those of others. Hence I here only allude to my 

 own attempted solution of the general equation of the fifth 

 degree (see Phil. Mag. S. 3, vol. xxvii. pp. 125, 126, 292, 

 293, vol. xxviii. pp. 190, 191), the failure of which I after- 

 wards (lb. vol. xxviii. p. 395) pointed out, and, subsequently 

 (see par IV. pp. 52, 53, vol. xxxii.), more clearly explained. 



25. In Sir W. R. Hamilton's discussion (referred to above) 

 of Mr. Jerrard's process, certain limits are given within which 

 the application of the method is confined. I have sought (see 

 Phil. Mag. S. 3. vol. xxix. pp. 181-183, vol. xxx. pp. 28-30) 

 to give a similar limitation to the processes of the method of 

 vanishing groups. But the results which I have arrived at 

 are not altogether satisfactory, and I should wish that any 

 judgement, formed on the subject, may be suspended until I 

 shall have been enabled to give a complete discussion of the 

 question, or, until some one, who may take a sufficient interest 



