New Mode of Resolving Algebraic Equaiiotis. S83 



to destroy belongs to the Creator alone, I entirely coincide 

 with Roget and Faraday in the opinion, that any theory which, 

 when carried out, demands the annihilation of force, is neces- 

 sarily erroneous. The principles, however, which I have ad- 

 vanced in this paper are free from this difficulty. From them 

 we may infer that the steam, while expanding in the cylinder, 

 loses heat in quantity exactly proportional to the mechanical 

 force which it communicates by means of the piston ; and that 

 on the condensation of the steam, the heat thus converted into 

 power is not given back. Supposing no loss of heat by radia- 

 tion, &c., the theory here advanced demands that the heat 

 given out in the condenser shall be less than that communi- 

 cated to the boiler from the furnace, in exact proportion to 

 the equivalent of mechanical power developed. 



It would lengthen this paper to an undue extent were I now 

 to introduce any direct proofs of these views, had I even lei- 

 sure at present to make the experiments requisite for the pur- 

 pose ; I shall therefore reserve the further discussion of this 

 interesting subject for a future communication, which I hope 

 to have the honour of presenting to the Royal Society at no 

 distant period. 



Oak Field, near Manchester, June 1844. 



LV. Outline of a New and General Mode of Transforming 

 and Resolving Algebraic TLquations, By James Cockle, 

 jB.^., of the Middle Temple, Special Pleader^. 

 1. nPHE practical application of the following will be found 

 -*• in various papers which I have had the honour of 

 publishing in the Mathematician. The method is, however, 

 here presented in an entirely novel form. Considered gene- 

 rally, its characteristic is, the effecting the proposed reduc- 

 tions by modifying the roots of an equation directly. By way 

 of commencement, I have, for this purpose, generalised the 

 assumption of Mr. Murphyf (which is undoubtedly true for 

 equations of the first four degrees), and assumed that the 

 roots of the general equation of the nlh degree, in j/, are given 

 by a set of expressions of which the type is 



i/, = /3o+«^/3i + «2'-/32+ . . . +a('^-i>/3„_i, . (1.) 



where a denotes one of the wth roots of unity. It follows 

 from this, that 



J/i + ai/2 + «'i/3 +....+ a'^'X = ^^„_i ; • (2.) 



and, denoting the left-hand side of (2.) by <^ {y), if <p {y) = 0, 



/3«_i=0. 



* Communicated by T. S. Davies, Esq., F.R.S., F.S.A. 

 t Philosophical Transactions, 1 837, part I. 



