Mr. Rankine on the Stability of Earthwork and Masonry, 469 



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

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

 of natural slope of the earth, 



6(1 07 TI. Principle of the Transformation of Structures. 



''"Lei 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 

 SL 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- 

 trados), and in which ok=OKy oa=OA, and a6=AB, are the same 

 as in the primitive arch fig. 1 . 







/aainBiLo'iff; 



Solution. On any vertical plane passing through BK, and not 

 coinciding with the plane of fig. 1, draw cod of the given length and 

 incHnation, intersecting COD in O. Join Ce, Dd, and project the 



