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



for the equiUbrium of a rigid parallelepiped, and will be shown in this memoir to 

 be satisfied by the equations of equilibrium of bodies whose molecules attract and 

 repel each other in the direction of the line joining them ; but if no supposition be 

 made as to the nature of the molecular action,there will be no condition resulting 

 from the equilibrium of couples ; for we have no right to assume, in the equili- 

 brium of a parallelepiped, whose elements may alter their relative position, and 

 thus develope new forces, that the same equations hold as in the equilibrium of 

 an isolated ric^id parallelepiped, for which case six equations of condition are 

 necessary, arising from the equilibrium of forces and of couples. 



Restoring the usual signification of {x, y, z) in equations (8), and omitting 

 the external forces, we obtain 



^d'-^^dP, dP, ^ dP, 

 dt^ ~ da; dy dz 



_ d?r}^_dA^ dQi dQi. /jQ\ 



^ dt- ~ dx dy dz ' 



dX _dR, dR^ dR, 



di^ ~ dx dy dz 



These equations correspond with (5), and by comparing them we may deduce 

 the following relations : 



dV_p dV_ dV_ 



^-^" da,~ " da,--"'' 



dV_^ dV_ dV 



dV p dV_ dV_ 



d^r " d^r-' wr '' 



from which we obtain the following theorem : 



" If through any point (x, y, z), three elements of planes be drawn parallel 

 to the co-ordinate planes, the total action of the part of the body lying at one 

 side of these planes will consist of three normal and six tangential forces ; and 

 these forces may be expressed by the difierential coefficients of the function V, 

 with respect to (a,, a,, a^, &c.) " This theorem is of importance in classifying 



