70 THE EQUILIBRIUM OF ELASTIC SOLIDS. 



Let R denote the radius of curvature of the curve of compression at any 

 point, and let S denote the length of the curve of dilatation at the same 

 point, 



and since (q p), R and S are known, and since at the surface, where <, (a;, y) = 0, 

 p = 0, all the data are given for determining the absolute value of p by inte- 

 gration. 



Though this is the best method of finding p and q by graphic construc- 

 tion, it is much better, when the equations of the curves have been found, that 

 is, when <, and <, are known, to resolve the pressures in the direction of the 

 axes. 



The new quantities are p lf p t , and q t ; and the equations are 



Pi~ Pi 



It is therefore possible to find the pressures from the curves of equal tint 

 and equal inclination, in any case in which it may be required. In the mean- 

 time the curves of Figs. 2, 3, 4 shew the correctness of Sir John Herschell's 

 ingenious explanation of the phenomena of heated and unannealed glass. 



NOTE A. 



As the mathematical laws of compressions and pressures have been very thoroughly 

 investigated, and as they are demonstrated with great elegance in the very complete and 

 elaborate memoir of MM. Lame" and Clapeyron, I shall state as briefly as possible their results. 



Let a solid be subjected to compressions or pressures of any kind, then, if through any 

 point in the solid lines be drawn whose lengths, measured from the given point, are pro- 

 portional to the compression or pressure at the point resolved in the directions in which the 

 lines are drawn, the extremities of such lines will be in the surface of an ellipsoid, whose 

 centre is the given point. 



The properties of the system of compressions or pressures may be deduced from those 

 of the ellipsoid. 



