AND THE EQUILIBRIUM ANU MOTION OF ELASTIC SOLIDS. 307 



ft)', o)" and w" denoting here the same as in Art. 2, and wa &c. denoting the initial values of 

 w, &c. for the same particle. Solving the above equations with respect to <«', u>" and u>"', the 

 resulting equations are 



1 idx , dx „ dx 



'"=s[d-a'"^ ^db"'" *Tc'"" y^^-' 

 where .S' is a function of the differential coefficients of x, y and z with respect to o, b and c, 

 which by the condition of continuitv is shown to be equal to — , n being the initial density about 



the particle whose density at the time considered is p. Since — &c. are finite, (for to suppose 



them infinite would be equivalent to supposing a discontinuity to exist in the fluid,) it follows at 

 once from the preceding equations that if &)„' = 0, at" - 0, w^" = 0, that is if u^da + v^db + w^dc 

 be an exact differential, either for the whole fluid or for any portion of it, then shall w = 0, 

 (u" = 0, (,,"' = 0, i.e. iid,v + vdy + wdz will be an exact differential, at any subsequent time, 

 eitlier for the whole mass or for the above portion of it. 



12. It is not from seeing the smallest flaw in M. Cauchy's proof that I propose a new one, 

 but because it is well to view the subject in different lights, and because the proof which I am 

 about to give does not require such long eciuations. It will be necessary in the first place to prove 

 the following lemma. 



Lemm.^. If o)|, w.,;...ai„ are n functions of t, which satisfy the n differential equations 



(25) 



at 



d,w„ 



-- = P„W, + Q„U>„... + V„w„, 



(if 



where P„ Q,... V„ may be functions of t, <«i...a.,„ and if when oj, = o, tu^ = 0...w„ = 0, none of the 

 quantities P,, ...V„ is infinite for any value of t from to T, and if (Oi...w„ are each zero when 

 f = 0, then shall each of these quantities remain zero for all values oft from to T. 



Demonstration. Let r be a finite value of t, then by hypothesis t may be taken so small 

 that the values of Wi...w„ are sufficiently small to exclude all values which might render any one of 

 the quantities P,...F„ infinite. Let Z, be a superior limit to the numerical values of the 

 several quantities Pi...V„ for all values of t from to t ; then it is evident that (o,...a)„ cannot 

 increase faster than if they satisfied the equations 



dwi 

 — = Z,((U, + (Uj ... + w„), \ 



—J = L(w, + (Uj,... + w„), j 



.(26) 



vanishing in this case also when f = 0. But if w, + w,... + w„ = il, we have by adding together 

 the above equations 



if now Q be not equal to zero, dividing this equation by <2 and integrating, we have 



il = Ce"-'; 



R R 2 



