AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 293 



ones, in which case the molecule in question would start into a new position of equilibrium. This 

 start would cause a corresponding displacement in the molecules immediately about the one which 

 started, and this disturbance would be propagated immediately in all directions, the nature of the 

 displacement however being different in different directions, and would soon become insensible. 

 During the continuance of this disturbance, the pressure on a small plane drawn through the 

 element considered would not be the same in all directions, nor normal to the plane : or, which 

 comes to the same, we may suppose a uniform normal pressure p to act, together with a normal 

 pressure p^^ and a tangential force < , p^^ and t^^ being forces of great intensity and short duration, 

 that is being of the nature of impulsive forces. As the number of molecules comprised in the 

 element considered has been supposed extremely great, we may take a time t so short that all 

 summations with respect to such intervals of time may be replaced without sensible error by 

 integrations, and yet so long that a very great number of starts shall take place in it. 

 Consequently we have only to consider the average effect of such starts, and moreover we may 

 without sensible error replace the impulsive forces such as p^^ and t^., which succeed one another 

 with great rapidity, by continuous forces. For planes perpendicular to the axes of extension 

 these continuous forces will be the normal pressures p', p", p'". 



Let us now consider a motion of shifting differing from the former only in having e' increased 

 in the ratio of m to 1. Then, if we suppose each start completed before the starts which would 

 be sensibly affected by it are begun, it is clear that the same series of starts will take place in the 

 second case as in the first, but at intervals of time which are less in the ratio of 1 to m. 

 Consequently the continuous pressures by which the impulsive actions due to these starts may be 

 replaced must be increased in the ratio of m to 1. Hence the pressures p', p", p" must be 

 proportional to e , or we must have 



p =Ce, p =Ce, p =Ce. 



It is natural to suppose that these formulae held good for negative as well as positive values 

 of e. Assuming this to be true, let the sign of e be changed. This comes to interchanging 

 X and y, and consequently p'" must remain the same, and p' and p" must be interchanged. We 

 must therefore have C" = 0, C = — C. Putting then C = - 2/ti we have 



p = - Zfie, p" = 2;u.e', p" = 0. 



It has hitherto been supposed that the molecules of a fluid are in actual contact. We 

 have every reason to suppose that this is not the case. But precisely the same reasoning will apply 

 if they are separated by intervals as great as we please compared with their magnitudes, provided 

 only we suppose the force of restitution called into play by a small displacement of any one 

 molecule to be very great. 



Let us now take the case of two motions of shifting which coexist, and let us suppose 

 e = <j + a', e" = - a, e" = - or'. Let the small time t be divided into 2w equal portions, and 

 let us suppose that in the first interval a shifting motion corresponding to e = 2a, e"= — 2a takes 

 place parallel to the plane x^ y., and that in the second interval a shifting motion corresponding 

 to e'= 2y', e"' = -2a-' takes place parallel to the plane at^ z^, and so on alternately. On this 



supposition it is clear that if we suppose the time — to be extremely small, the continuous forces 



by which the effect of the starts may be replaced will be p'= - 2/i((t + a), p "= 2;u(t, p ' = 2^0-'. By 

 supposing « indefinitely increased, we may make the motion considered approach as near as we 

 please to that in which the two motions of shifting coexist ; but we are not at liberty to do so, 



•j- 

 for in order to apply the above reasoning we nuist suppose the time — to be so large that the 



average effect of the starts which occur in it may be taken. Consequently it must be taken as an 

 additional assumption, and not a matter of al)solute demonstration, that the effects of the two 

 motions of shifting are superimposed. 



