2 REPORT 1866. 



Description of a New Proportion-Table, equivalent to a Slidiwj-rule 13 feet 

 4 inches lon<j. By J. D. Eteeett, D.C.L., Assistant Professor of Mathe- 

 matics in Glasgow University. 



The distinctive feature of the new arrangement consists in breaking np each of 

 the two pieces which compose a sliding-rule into a number of equal parts, and ar- 

 ranoing these consecutiveh' in parallel colmnns, the columns in one of the two pieces 

 beino- visible through openings cut between the columns of the other. The large- 

 ness of the scale is such that the space from 1 to 1-1 is divided into a hundred parts, 

 the smallest of these being ^-V of an inch long. The material employed is Bristol 

 board, printed from copper plates, the dimensions of each board, exclusive of mar- 

 gin, being 16 by 8 inches. 



Multiplication and division can be performed by this Table with the same ac- 

 curacy as by fom'-figau-e logarithms, and with greater ease and expedition, Formulie 

 not aiiapted to logarithmic computation are thus rendered available, and with the 

 aid of a small table of natural sines and tangents the calculations of nautical astro- 

 nomy can be performed with great facility and with all needful accm-acy. 



On certain Errors in the received Equivalent of the Metre, Sfc. By F. P. Fellows. 



On Tschirnhausen's Method of Transformation of Algehraic Equations, and 

 some of its Modern Extensions. By the Rev. Prof. E. Haelet, F.E.S. 



It has long been known that any algebraic equation may be deprived of its second 

 term by a linear transformation. Tschirnhauseu introduced quadratic, and suggested 

 higher'transfomiations, and thus opened the way to great progi-ess in the theory. 

 He showed that by the solution of a linear equation and of a quadi-atic, any alge- 

 braic equation may be deprived of its second and third terms siuudtaneously. The 

 complete quintic 'may in this way be reduced to a quadrinomial form. Erland 

 Bring, Professor of History in the University of Lund, in a paper bearing date 14th 

 December 178G, seems to have been the first to extend Tscliirnhausen's method so 

 as to reduce the quintic to a trinomial form by depriving it of its second, third, and 

 fourth terms simultaneously. (See a paper by Prof Harlev, entitled " A Contri- 

 bution to the History of the Problem of the Reduction of the General Equation of 

 the Fifth Degree to a Trinomial Form," Quarterly Jom'nal of Mathematics, vol. vi. 

 pp. 38-47.) Bring's process has lately been simplified by ^Mr. Samuel Bills of 

 Hawton, near Newark ; and Prof Ilarley explained to the Section how Mr. Bills's 

 method might be extended so as to deprive the general equation of the «th degTee 

 of its second, third, and fourth terms by the solution of equations none of which 

 rise higher than the third degree ; and of its second, third, and fifth terms by the 

 solution of equations none of which rise higher than the fourth degi-ee. (See 

 " Mathematics fi-om the Educational Times," vol. i. pp. 8, 38-40, 57, 58.) Notice 

 was taken of the labours of other in^'estigators in the same field, particularly 

 Mr. Jerrard, Sir W. R. Hamilton, Chief Justice Coclde (Queensland), Prof Cayley, 

 and Prof Sylvester. The author concluded with some observations on the alleged 

 solutions of the general quintic by the late ]SL\ Jerrard and Judge Hargreave. 



On Differential Resolvents. By the Rev. Professor R. Haeley, F.R.S. 



The author gave a short accoimt of his researches on difierential resolvents, 

 particularly those connected with certain trinomial forms of algebraic equations, 

 .(in abstract of these researclies has recently been published by the Loudon Mathe- 

 matical Society. He also pointed out the coincidence of some of his own results 

 with those obtained about the same time, quite independently, by Chief Justice 

 Cockle, F.R.S., of Queensland. 



The difierential equation 



7 r d'-\n „rn — r d ,m -."[n-r r-f d m -,'!''„ 



6r a: I u=a" \ x- — |- — — 1 -.r- — — 1 ,r"« 



L a.rj i_ n a.v n J Lh dx n J 



(in which the ordinary factorial notation 



'm' =^0-1) (^-2) . . . (v-e+i) 



is adopted) is satisfied by the with power of any root of the algebraic equation 



