106 Oil the Equation of Continuity in Fluid Motion. 



. „ Jdvo^ dw^ div \ (dv^ , dv 



•^ "Xdx dy '' dz J \dx dy 



=^-^^Hr>Ty-^T:\ 



. . -, rdu ^y du 82; dv '6x dv h z dw 8a? 

 + 6a;.bj/.6s.|_— .^ + — .g- + — . — + — .g^ + ^.g^ 



dii) 8i/~l 

 + ^*8iJ* 



dx dy dz ' 



we have Dt(pY) = 0, or p.DjV + V.D</5=0; 



D.p=/^ + u. 



or 



y J 5, /du dv dw\ , 5, s, ^ /du 8w J?< 8s \ 



p.ox.hj.SzA — + — + —- ) + p. 8a?. 8?/. 8^. ( -r- . TT^ + -7-.^ + &c. I 

 "^ -^ Xdx^dy^d^J ^ -^ Vrf?/ 8a?^rf.^8x / 



H-S.,..8,.^^-£+„.g + „.|+».£)=0. 



Now since there is no necessary connexion between 8.r, ly 

 and 8^, the middle term in this equation, viz. that containing 



the quantities ^, &c., cannot affect the other two : according 



therefore to a well-known principle, the two parts must sepa- 

 rately = zero. 



Therefore, finally, dividing out by the common factor 8a\ 

 iy.^z, we get the usual equation of continuity, 



(du dv dxo\ dp dp dp dp 



Tx + d^ + Tz)+dt-^"-dx + ''-i + '"-dz='''' 

 dp djpu) djpv) djpw) _ 

 dt^ dx + dy + dz 

 It is obvious that this equation only requires that the nnm- 

 ber of particles in the element should remain unaltered during 

 [dt) ; but the mode in which it is obtained implies, not only an 

 identity in the number of particles at the beginning and end of 

 {dt), but an identity of the pat-tides themselves. This obser- 

 vation has also been made by Poisson, whose words are (art. 

 651), "c'est pour abreger que I'on a considere le volume de 

 cette partie du fluide conime infiniment petit; et si Ton divise 



