AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 297 



to denote differentiation in wliicli the independent variables are t and three parameters of the 



particle considered, (such for instance as its initial co-ordinates,) and not t, ,v, y, «. It is easy also 



to show that the moving force acting on the element considered arising from the oblique pressures 



, „ . , . , (dP dTs dT.,\ 

 on the faces is ultimately ( - — + —— + -— 1 Aa? Ay Aar, acting in the negative direction. Hence 



we have by D''Alembert's principle 



(Du „\ dP, dT-i dT, 



lUu 



"^ -T + ~r -^ ^r = 0'&c-' (10) 



ax ay dz ' 



.... . Du . , du du du du Dv 



in which equations we must put tor =— its value -— + u + v + w — , and siniilarlv for — 



' Dt dt da; dy dz ^ dt 



and — — • In considering the general equations of motion it will be needless to write down more 



than one, since the other two may be at once derived from it by interchanging the requisite 

 quantities. The equations (lO), the ordinary equation of continuity, as it is called, 



dp dpu dpv dpw 



:£ + -; +^ + ^5^ = (11) 



dt dx dy dz '■ 



which expresses the condition that there is no generation or destruction of mass in the interior 

 of a fluid, the equation connecting p and p, or in the case of an incompressible fluid the equivalent 



equation _- = 0, and the equation for the propagation of heat, if we choose to take account 



of that propagation, are the only equations to be satisfied at every point of the interior of 

 the fluid mass. 



As it is quite useless to consider cases of the utmost degree of generality, I shall suppose 

 the fluid to be homogeneous, and of a uniform temperature throughout, except in so far as the 

 temperature may be raised by sudden compression in the case of small vibrations. Hence in 

 equations (10) fi. may be supposed to be constant as far as regards the temperature; for, in the 

 case of small vibrations, the terms introduced by supposing it to vary with the temperature 

 would involve the square of the velocity, which is supposed to be neglected. If we suppose 

 fj. to be independent of the pressure also, and substitute in (lO) the values of P, &c. given by (8), 

 the former equations become 



fDu \ dp id-u d'u d'u\ /j. d idu dv dw\ 



p[m-^)-'d:r-''\d^^'-df'-d^)-ld:v[d7v^Ty^d-z) = ''^ ^' <''> 



Let us now consider in what cases it is allowable to suppose n to be independent of the 

 pressure. It has been concluded by Dubuat, from his experiments on the motion of water in 

 pipes and canals, that the total retardation of the velocity due to friction is not increased by 

 increasing the pressure. The total retardation depends, partly on the friction of the water 

 against the sides of the pipe or canal, and partly on the mutual friction, or tangential action, 

 of the different portions of the water. Now if these two parts of the whole retardation were 

 separately variable with p, it is very unlikely that they should when combined give a result 

 independent of p. The amount of the internal friction of the water depends on the value of ,1*. 

 I shall therefore suppose that for water, and by analogy for other incompressible fluids, « is 

 independent of the pressure. On this supposition, we have from equations (II) and (12) 

 fDu \ dp ld?u d^M d^u\ 



'^ l^- ^) ^ rf^ - " (rf^^rf? ^ rf^j = °' ^^ ('^> 



du dv dw 



dx dy dz 

 Vol.. VIII. Paut hi Qq 



