184 The Rev. J. H. Jellett on the Equilibrium and Motion of an Elastic Solid. 



which represents the force which m' exerts upon n* in the original position of 

 these particles, and 



^l(^'-?) + £(,/-,;) + G(r-f), 



the force generated by the displacement. 



The supposition that the original state of the body was one of free equili- 

 brium permits us to disregard the former of these parts. For it follows from 

 that supposition, that if the several particles ofthe body receive equal displace- 

 ments, the new position is also a position of equilibrium. Hence the suppositions, 



r=«, n'=% r=r, 



must satisfy the general equation of equilibrium. But these suppositions give 



Hence the terms depending upon Fo will disappear of themselves. We have, 

 therefore, for the effective part of the force, 



F= A {^ -l;) + B (,/ - ,,) -f C (r - .0, 



where ^ , JS, C are in general of the form 



/ {x, y, z, p, e, 4>). 



Let a, /3, 7 be the angles which the direction of p makes with the axes, so 



that 



cos a — sin cos 0, cos ^ — sin 6 sin 0, cos y = cos 6. 



Then since 



dx — p cos a, dy = p cos ^, dz = p cos 7, 



we shall have from equations (A), 



I dk ^ di, r/A 



^'_^^p(^C0Sa-^ + C0S^^ + C0S7^.j, 



^'-^=P^cosa^ + cos^^ + cos7'£j, (C) 



( di: ^dr dt:\ 



r-f=p(^cosa^ + cos^^ + cos7^J. 



and therefore, 



