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X. On Hamilton's Numbers. 
By J. J. Sylvester, D.O.L., F.R.S., 
Savilian Professor of Geometry in the University of Oxford, 
and James Hammond, M.A. Cant. 
Received June 11,—Read June 16, 1887. 
Introduction. 
In the year 1786 Erland Samuel Bring, Professor at the University of Lund in 
Sweden, showed how by an extension of 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 ascribed by him to one of his 
own pupils, a certain Sven Gustae Sommelius, and embodied in a thesis humbly 
submitted to himself for approval by that pupil, as a preliminary to his obtaining his 
degree of Doctor of Philosophy in the University." 1 ' The process for effecting this 
reduction seems to have been overlooked or forgotten, and was subsequently re-dis¬ 
covered 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 
unnecessary refutation of a result, known a, priori to be impossible, by an extremely 
valuable 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 
* Being’s “ Reduction of the Quintic Equation ” was republished by the Rev. Robeet Haeley, 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. Haeley (see ‘ Quart. Journ. Math.,’ pp. 44, 45) is as follows: “ B. cum D. 
Meletemata quaedam mathematica circa transformationem aequationum algebraicaram, quae consent. 
Ampliss. Pacult. Philos, in Regia Academia Carolina Praeside B. Eeland Sam. Being, Hist. Profess. 
Reg. & Ord. publico Eruditorum Examini modeste subjicit Sven Gustaf Sommelius, Stipendiarius Regius 
& Palmcrentzianus Lundensis. Die xiv Decemb., mdcclxxxyi, L.H.Q.S.—Lundae, typis Berlingianis.” 
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