Designing by Geometry. Rankine's Theorems of Transformation of Structures.

William John Macquorn Rankine (1820-1872) was one of the main figures in establishing engineering science in the second half of the 19th. Century. His Manual of Applied Mechanics (1858) gathers most of his contributions to strength of materials and structural theory. A few additions are to be found in his Manual of Civil Engineering (1862). The book is based in his Lectures on Engineering delivered in the Glasgow University, and formed part of his intention of converting engineering science in a university degree (Channell 1982, Buchanan 1985). Both in plan and in content the book shows and enormous rigour and originality. It is difficult to read. As remarked by Timoshenko (1953, 198): "In his work Rankine prefers to treat each problem first in its most general form and only later does he consider various particular cases which may be of some practical interest. Rankine's adoption of this method of writing makes his books difficult to read, and they demand considerable concentration of the reader." Besides, Rankine does not repeat any demonstration or formula, and sometimes the reader must trace back the complete development through four or five previous paragraphs. The method is that of a mathematician. However, the Manual had 21 editions (the last in 1921) an exerted a considerable influence both in England and America. In this article we will concentrate only in one of the more originals contributions of Rankine in the field of structural theory, his Theorems of Transformation of Structures. These theorems have deserved no attention either to his contemporaries or to modern historians of structural theory. It appears that the only exception is Timoshenko (1953,198-200) who cited the general statement and described briefly its applications to arches. The present author has studied the application of the Theorems to masonry structures (Huerta and Aroca 1989; Huerta 1990, 2004, 2007). Rankine discovered the Theorems during the preparation of his Lectures for his Chair of Engineering in the University of Glasgow . He considered it very important, as he published it in a short note communicated to the Royal Society in 1856 (Rankine 1856). He included it, also, in his article "Mechanics (applied)" for the 8th edition of the Encyclopaedia Britannica (Rankine 1857). Eventually, the Theorems were incoroporated in the Manual of applied mechanics and applied to frames, cables, rib arches and masonry structures. The theorems were also included in his Manual of civil engineering (1862), generally in a shortened way, but with some additions

Multiplicative excellent families of elliptic surfaces of type E_7 or E_8

We describe explicit multiplicative excellent families of rational elliptic surfaces with Galois group isomorphic to the Weyl group of the root lattices E_7 or E_8. The Weierstrass coefficients of each family are related by an invertible polynomial transformation to the generators of the multiplicative invariant ring of the associated Weyl group, given by the fundamental characters of the corresponding Lie group. As an application, we give examples of elliptic surfaces with multiplicative reduction and all sections defined over Q for most of the entries of fiber configurations and Mordell-Weil lattices in [Oguiso-Shioda '91], as well as examples of explicit polynomials with Galois group W(E_7) or W(E_8).Comment: 23 pages. Final versio