106 The Rev. S. Haughton on a Classification of Elastic Media, 



of the body situated at opposite sides as acting upon any element of the plane ; 

 in general, the effect of the particles at one side will be to produce a normal 

 force, and tangential forces acting in the element ; these tangential forces may 

 be resolved into two directions at right angles to each other. Let us now con- 

 ceive an elementary parallelepiped, with one corner situated at the point {x,y,z) 

 (these co-ordinates here denote the actual position of the molecule at any in- 

 stant) ; let the forces acting on the side (dydz) be denoted by {P^, Q„ i?,), P, 

 being normal and parallel to the axis of (.■;■), (Q,, /?,) tangential and parallel to 

 the axes of Y and Z ; let the forces acting on the side {dxdz) be (Pa, Qi, R-i), 

 Qi being the normal force ; and let the forces at the side {dxdy) be (P3.Q3.-ffs). 

 P3 being the normal force. The forces acting on the opposite sides of the 

 parallelepiped will be : — on the side opposite to (dydz), 



h+^dji\dydz, fQ,+^dw)dydz, (jR, +^ dx\ dydz ; 

 on the side opposite to (dxdz), 



(P2 + -r^dy]d.rdz, (Qi + -,— dij\dxdz, (n.+'-pdyjdxdz; 

 on the side opposite to (dxdy), 



fp^ + '^dz]dxdy, (Q, + ^dz]dxdy, (B, + ^dz)dxdy. 



These forces, acting on the six sides of the parallelepiped, must equilibrate the 

 forces (Xdm, Ydm, Zdin), applied at the centre of the parallelepiped, and 

 arising from external causes ; hence, the equations of equilibrium will be 



(dP, dP, dP,\ , , , ^ 

 -r^j fdQ, dQ. dQ,\ , , , 



^'""^^[-d^^^^-ir)''^^'^'-'' 



„, fdRi dRi dR^ , 7 J ri 

 Zdm + (-7- +"j-^ +-;r~ ) dxdydz = 0; 



or, since pdxdydz = dm, and the molecular forces must act in the direction op- 

 posite to the applied forces, including negative accelerating forces, 



