458 Mb. HOPKINS, ON THE INTERNAL PRESSURE OF ROCK MASSES. 



Similarly, the couple arising from the forces B' and B' + -j^ Sx about the same axi.s parallel to z, 



will be 



— Bcaidydsi. 



Also the moment of the normal forces A, B, C, with reference to the above-mentioned axes, 

 will be zero, always omitting small quantities of tlie fourth order. Consequently the whole moment 

 of the forces on the parallelopiped with reference to the axis parallel to that of z, will be 



{J' -B')S:^v^ySz; 

 which must = zero by the conditions of equilibrium ; and therefore 



J' = B'. 

 In exactly the same way we find, by taking the moments with reference to the axes parallel 

 respectively to these of y and oo, 



A" = C, 



B" = C". 



By means of these three relations the six accented quantities are reduced to three independent 

 quantities. 



2. Let us now conceive a plane to meet the three co-ordinate planes so as to form with them 

 a tetrahedron, whose vertex is at the origin P. Suppose the exterior normals to the three faces 

 formed by the co-ordinate planes to point respectively towards the positive directions of x y and 

 z ; and let a ji and 7 be the angles which the normal to the base of the tretrahedron makes 

 with the co-ordinate axes of .r y and z. Also let s denote the area of the base, and s' s" and s" 

 the areas of the sides of the tetrahedron perpendicular respectively to the axes of ,r y and z, all 

 these quantities being indefinitely small. 



Again, let ps denote the whole resultant force acting on *■, and let \ in and v be the angles 

 which its direction makes with lines parallel to the co-ordinate axes of w y and z, this direction being 

 exterior to the tetrahedron. Then, in order that the tetrahedron may be in equilibrium, we nnist 



have 



ps . cos X = As' + A's" + A"s"', 



ps .cos n = Bs" + B's + B"s"', 

 s + L s + L s \ 



but 



s s s 



— = COS a, — = cos fi, — = cos 7 ; 



s s s 



making these substitutions, and also putting 



B" = C" = D, 



A" = C = £, 



A' = B' = F, 



we shall have 



p . cos \ = A cos a + F cos (i + E cos 7, \ 



p . cos fk— B cos fi + F cos a + D cos 7, \ (a), 



p . cos v = C cos y + E cos a + D cos /3 ; | 



formulae in which the notation agrees with that of M. Cauchy (Exercises de Mathematiqut, 

 Vol. 11. p. 48). 



