470 



Messrs. J. J. Sylvester and J. Hammond. [June 16, 



XXVIII. "On Hamilton's Numbers." By J. J. Sylvester, 

 F.R.S., Savilian Professor of Geometry in the University of 

 Oxford, and James Hammond, M.A. Cant. Received June 11, 

 1887. 



(Abstract.) 



In the year 1786 Erland Samuel Bring, Professor at the University 

 of Lund in Sweden, discovered that by the method of Tschirnhausen 

 it was possible to deprive the general algebraical equation of the 5th 

 degree of three of its terms without solving an equation higher than 

 the 3rd degree. By a well understood, however singular, academical 

 fiction, this discovery was imputed by him to one of his own pupils, 

 one Sven Grustaf Sommeliiis, and embodied in a thesis humbly sub- 

 mitted to himself for approval by that pupil, as a preliminary to his 

 obtaining his degree of Doctor of Philosophy in the University.* It 

 seems to have been overlooked or forgotten, and was subsequently 

 re-discovered many years later by Mr. Jerrard. In a report contained 

 in the ' Proceedings of the British Association ' for 1836, Sir William 

 Hamilton showed that Mr. Jerrard was mistaken in supposing that 

 the method was adequate to taking away more than three terms of the 

 equation of the 5th degree, but supplemented this somewhat unneces- 

 sary refutation by a profound and original discussion of a question 

 raised by Mr. Jerrard, as to the number of variables required in 

 order that any system of equations of given degrees in those variables 

 shall admit of being satisfied without solving any equation of a 

 degree higher than the highest of the given degrees. 



In the year 1886 the senior author of this memoir showed in a 

 paper in Kronecker's (better known as Crelle's) ' Journal ' that 

 the trinomial equation of the 5th degree, upon which by Bring's 

 method the general equation of that degree can be made to depend, has 

 necessarily imaginary coefficients except in the case where four of the 

 roots of the original equation are imaginary, and also pointed out a 

 method of obtaining the absolute minimum degree M of an equation 

 from which any given number of specified terms can be taken away 

 subject to the condition of not having to solve any equation of a 



* Bring's Eeduction of the Quintic Equation was republished by Mr. Robert 

 Harley, F.R.S., in the ' Quarterly Journal of Pure and Applied Mathematics,' vol. 6, 

 1864, p. 45. The full title of the Lund Thesis, as given by Mr. Harley (see 

 ' Quart. Journ. Math.,' pp. 44, 45) is as follows : " B. cum D. Meletemata quaedam 

 mathematica circa transformationem aequationum algebraicarum, quae consent. 

 Ampliss. Facult. Philos. in Regia Academia Carolina Praeside D. Erland Sam. 

 Bring, Hist. Profess. Reg. & Ord. publico Eruditorum Examini modeste subjicit 

 Sven Gustaf Sommelius, Stipendiarius Regius & Palmcrentzianus Lundensis. 

 Die xiv Decemb., mdcclxxxvi, L.H.Q.S. — Lundae, typis Berlingianis." 



