AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 311 



there may also, in the case of motion, be a small part depending on the relative velocities, 

 but this part may be neglected, unless we have occasion to consider the effect of the internal 

 friction in causing the vibrations of solid bodies to subside. It has been shown (Art. 7.) that 

 the effect of this cause is insensible in the case of sound propagated through air; and there 

 is no reason to suppose it greater in the case of solids than in the case of fluids, but rather 

 the contrary. The capability which solids possess of being put into a state of isochronous 

 vibration shows that the pressures called into action by small displacements depend on homo- 

 geneous functions of those displacements of one dimension. I shall suppose moreover, according 

 to the general principle of the superposition of small quantities, that the pressures due to 

 different displacements are superimposed, and consequently that the pressures are linear functions 

 of the displacements. Since squares of a, /3 and y are neglected, these pressures may be referred 

 to a unit of surface in the natural state or after displacement indifferently, and a pressure which 

 is normal to any surface after displacement may be regarded as normal to the original position 

 of that surface. Let - ^^ be the pressure corresponding to a uniform linear dilatation ^ when 

 the solid is in equilibrium, and suppose that it becomes - mJS, in consequence of the heat 

 developed, when the solid is in a state of rapid vibration. Suppose also that a displacement 

 of shifting parallel to the plane a;y, for which a = kx, /3 = - ky, 7 = 0, calls into action a 

 pressure - Bk on a plane perpendicular to the axis of a:, and a pressure Bk on a. plane 

 perpendicular to that of y; the pressures on these planes being equal and of opposite signs, 

 that on a plane perpendicular to the axis of ss being zero, and the tangential forces on those 

 planes being zero, for the same reasons as in Sect. 1. It may also be shown as before that 

 it is necessary to suppose B positive, in order that the equilibrium of the solid medium may 

 be stable, and it is easy to see that the same must be the case with A for the same reason. 



It is clear that we shall obtain the expressions for the pressures from those already found 

 for the case of a fluid by merely putting a, jS, 7, B for u, v, w, fx and -AS or -mAS for p, 

 according as we are considering the case of equilibrium or of vibratory motion, the body being in 

 the latter case supposed to be constrained only in so far as depends on the motion. 



For the case of equilibrium then we have from equations (8) 



,...., ..S(±L-S]. r,.-«(^.g),.o ,«, 



(da rf/3 dy\ 

 — + j~ "*" j~) ' ^""^ '•^^ equations of equilibrium will be obtained from (12) by 



Du 



putting -r— = 0, p = - AS, making the same substitution as before for m, v, w and u.. We have 



therefore, for the equations of equilibrium, 



^ ^ , , ^. ^ /<*« ^/3 dy\ „ id'u d'a d'a\ 

 , ^,1(^,5)- (- + -+£). 5 (^--.,.^^-.^^)=o,&c (30) 



In the case of a viliratory motion, when the body is in its natural state except so far as depends 

 on the motion, we have from equations (8) 



„ id-a „\ ^ „ idB dy 

 P,= -mAd-^2B [—-d], T,= - Bi-f- +-Zl,&c., (31) 



^dx I \dx dyf 



Du 

 and the equations of motion will be derived from (12) as before, only &c. must be replaced by 



d'a 



— &c., and X, Y, Z put equal to zero. The equations of n)otion, then, are 



d'a , , . n^ d Ida dfi dy\ ^ id'a d'a d'a\ 

 at ^ dx \dw dy dzl \dx' dy' d^'l 



