462 



Mr. HOPKINS, ON THE INTERNAL PRESSURE OF ROCK MASSES. 



5. If 7',, 7*2, and T^ be the corresponding values of the tangential force, we have 



r,^ = i (5 - cy, T.? = 1 (^ - cf, r/ = \{a- sy. 



If J, B, C be taken as they always may be, so that A shall be the greatest and C the least, 

 T.2 will be the greatest of these values, and I shall shew that it in fact is the only one which 

 satisfies the conditions of being a maximum. To do this, and to explain the relations of these 

 particular values of the tangential force to its general values, it will be convenient to have recourse 

 to a geometrical representation, analogous to that before spoken of with reference to the normal 

 forces. For this purpose assume 



p sin ^ = Tf, — , 

 r- 



w = r cos a, y = r cos /3, x = r cos y ; 



where T„ denotes a constant force, and c a constant line. Then equation (3) becomes 



r^V = Pa?y + Q'ai'z^ + R'y'z' (4), 



the equation to a surface such that the inverse of the square of the radius vector from the point P, 

 will be proportional to the tangential force on the plane s when perpendicular to that radius vector. 

 To find the intersections of the surface and the co-ordinate planes, put x — o, y — 0, and sr = 0, 

 consecutively ; we thus have 



cots = ^ C**, 



Q 



(Fig. 1.) 



^y= ± pC% 



as the equations to the intersections, each of which consists of two equal hyperbolas referred to the 

 asymptotes as axes of co-ordinates, as repre- 

 sented in the annexed diagram. PA, making 

 equal angles with the two co-ordinate axes in 

 the plane of the paper, is the semi-axis major, 

 and minimum radius vector in the hyperbola 

 whose vertex is A. Its position and that of 

 PA' correspond to the first, second, or third of 

 the systems of values (e) of a /3 and y, ac- 

 cording as the plane in which the hyperbolas 

 lie is that of yss, acx, or xy. Also the values 



of — -- in these cases respectively are 



R }_ Q 1 Pi 



2T"„V' iT„'"?' 27^V' 



which are proportional to 



R, Q, P; 



or to B -C, A -C, A-B, 



or to the three tangential forces previously 

 designated by 



T„ T,, n. 



