Mr. HOPKINS, ON THE INTERNAL PRESSURE OF ROCK MASSES. 459 



If o denote the angle between the direction of jo and the normal to s, we shall have p cos ^ 

 for the whole tiormal force acting on the area s in a direction exterior to the tetrahedron, and 

 p sin ^ the whole tangential force acting on the same area. Our first object will be to determine 

 a /3 and y, or the position of the base (s) of the tetrahedron, so that the normal action upon it, 

 p cos ^, shall be a maximum. We shall afterwards have a similar investigation with reference to 

 the tangential force p sin ^. 



We iiave cos ^ = cos X cos a + cos /i cos /3 + cos v cos y, 



whence we immed ately obtain 

 p cos I = A cos- a + B cos- /3 + C cos- y + 2D cos fi cos y + 2 £ cos a cos y + 2F cos a cos/3, ... (1), 



and since cos'' a + cos-/3 + cos'' 'y = 1, (2), 



we have {L being an arbitrary multiplier), 



(A + L) cos- a + (B + L) cos- (i + {C + L) cos^ y + 2 2) cos /3 cos y + I.E cos a cos y 



+ 2F cos a cos /3 = max. 

 Hence, 



\{A -y L) cos a + E cos y + F cos /3| sin a = J 



{ (fi + Z) cos /3 + Z» cos 7 + Z' cos a I sin /3 = ! (*)• 



{{C + L) cos 7 + i) cos /3 + £ cos a} sin 7 = ) 



To satisfy these equations together with 



cos' a + cos ' /3 + cos^ 7 = ] , 



we must equate the first brackets to zero. We thus have four equations from which L may be 

 eliminated, and « (i and 7 determined. 



If we multiply the first factors on the left-hand sides of equations (6) by cos a, cos /3 and cos 7 

 respectively, and add them together, we have by virtue of equations (a), 



L = — P cos ^, 

 and substituting for L in equations (6), we have 



p cos b cos a - A cos a + F cos (i + E cos 7, 

 = p cos X ; 



cos S cos a = cos X. 

 Similarly, cos ^ cos /3 = cos /i, 



cos c) cos 7 = cos V ; 

 whence cos"^ = 1, 



S = 0, 

 which shews that when the resultant force p is a maximum or minimum, its direction coincides 

 with that of the normal to the plane v. Consequently, also, the tangential force p sin S then 

 becomes = zero. 



This value of S gives, I. = — p, 



and sub.stituting for /, in equations (6) we have, 



(A - p) cos a + F cos fi + E cos 7 = 0) 



f cosa + (Z? - p) cos/3 + Z)cos7 = (")• 



E cos a + D cos (3 + {C - p) cos 7 = 



