
EQUILIBRIUM OF ELASTIC SOLIDS. 95 
doz See: sean 





dy rae 
dé Oz 
7-8 6,= a OG: = 4, Paci.) 
—_ a0y= = +30, == % 


By substituting in Equations (3.) the values of the forces given in Equa- 
tions (8.) and (9.), they become 
(+E a (= ee *)) + 3s (Fadetg dy + ea0) +pX=0 


(+s Ln) (= E (2 tt, 2)) 8 (Fer Z 209 +7 w302) + p¥ =0 (12.) 

(ut dm a) (Z(G aye 
These are the general equations of elasticity, and are identical with those 
of M. Caucny, in his Exercises d’ Analyse, vol. iii., p. 180, published in 1828, when 
2 
5 (fe det ee qb + gabe) + pZ=0 
k& stands for m, and K for ua and those of Mr Sroxes, given in the Cam- 
bridge Philosophical Transactions, vol. viii.’ part 3, and numbered (30.); in his 
equations A= 3p, B=> : 
If the temperature is variable from one part to another of the elastic solid, 
ae ae, 4 = at any point will be diminished by a quan- 


the compressions 
tity proportional to the temperature at that point. This principle is applied in 
Cases X. and XI. Equations (10.) then become 

dz a) (P+ Pot+Ps) +050 + = Py 
ne > i 
(Ga- Tau) (P+ Pet Ds) +040 at tas Nos) 
ae! 1 : e 
= (Fe Fu) (Pi+Ps +P, +0,0+— Dy 
c, v being the linear expansion for the temperature 2. 
Having found the general equations of the equilibrium of elastic solids, I 
proceed to work some examples of their application, which afford the means of 
determining the coefficients 4, m, and , and of calculating the stiffness of solid 
figures. I begin with those cases in which the elastic solid is a hollow cylinder 
exposed to given forces on the two concentric cylindric surfaces, and the two 
parallel terminating planes. 
VOL. XX. PART I. 2¢ 
