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II. " On the Mathematical Theory of the Stability of Earth- 

 work and Masonry." In a Letter to Prof. STOKES, Sec. 

 R.S. By W. J. MACQUORN RANKINE, Esq., C.E., F.R.S., 

 Professor of Civil Engineering in the University of Glasgow. 

 Received February 19, 1856. 



In the preparation of my course of lectures, I have found it neces- 

 sary to re-investigate much of the above-named branch of mechanics, 

 and I have now a paper in preparation on the subject, which I pro- 

 pose to offer to the Royal Society when it is ready. In the mean- 

 while, it appears to me that the two fundamental principles on which 

 my researches are based are of such a nature, that they may very 

 properly be communicated to the Royal Society at once. They are 

 as follows : 



I. Principle of the Stability of Earth. 



At each point in a mass of earth the directions of greatest and 

 least compressive stress are at right angles to each other ; and the 

 condition of stability is, that at each point the ratio of the diiference 

 of those stresses to their sum shall not exceed the sine of the angle 

 of natural slope of the earth. 



II. Principle of the Transformation of Structures. 

 Let a structure of a given uniform transverse section be stable 

 under a system of forces represented by given lines in the plane of 

 section : Then will any other structure whose transverse section is 

 a projection by parallel lines of that of the first structure upon any 

 other plane, be stable under the system of forces represented by the 

 projections, upon the new plane, of the lines representing the first 

 system of forces. 



Example of the application of this principle. 



Let fig. 1 be an equilibrated arch with its abutments of the form 

 (for example) proposed by M. Yvon-Villarceaux, suited for a hori- 

 zontal extrados EF. OK, OA, and AB being given, all the dimensions 

 of the arch and abutments are functions of those three quantities. 



It is required to design an arch, fig. 2, for an extrados ef, at any 

 given inclination, of any given span cd (measured parallel to the ex- 



