of Continuity in Fluid Motion. 105 



continuous, /. e. as changing their value continuously for amj 

 change of .r, y or z, then in order for X, Y, Z to be the same 

 for every part of an element, that element must be also con- 

 sidered a mathematical point, having no distinction of parts at 

 all. 



In nature, however, the forces are physicallij and not mathe- 

 matically continuous, and the elements on wliich they act may 

 have an infinite range of magnitude. The reasoning by which 

 such forces are submitted to calculation is however precisely 

 the same as if they were mathematically continuous. 



The element of fluid, then, may be of any finite magnitude 

 below a certain limit, and will consist of a number of ultimate 

 particles, on each of which the forces acting are the same. 

 For the whole extent of this element X, Y, Z are the same, 

 and it may be taken of any size consistent with this condition. 

 The next thing to be attended to is, that the element must 

 not be acted upon by any ne-w forces during the instant {dt). 

 During this interval, therefore, every particle in the element 

 IS acted upon solely by the forces X, Y, Z, which vary or 

 may vary with the time, but vary similarly for each of these 

 particles. In other words, there is no disturbing force intro- 

 duced during [d t) into the system of ultimate particles, which 

 together make up the element we are considering ; therefore 

 there can be no disturbance of the relative jjositions of these 

 particles. Therefore the numher of particles composing the 

 element remains the same during the interval {dt)\ and this 

 condition expressed analytically gives us the equation of con- 

 tinuity. 



Let |5 be the density of the element at the commencement 

 of the time {dt) ; this is supposed to be the same for the whole 

 of the element ; V the volume of the element at the beginning 

 of {dt) and V at the end. We have then (p' being also the 

 density at the end of the time {dt)) 



We have now merely to express this condition in another 

 form. The notation is the same as Poisson's. N = {x' — x){ij —y) 

 (z'—z): where for {.v'—x) we may write 8.r, for {y' — iy), iij, 

 and for {^ — z), 8 ^. We have to state in terms of the velocities 

 (m, V, w) parallel to the coordinate axes, that the variation of 

 {^^-a.'){y' — ij){z' — x)p is nothing during the instant (r//). Now 

 calling the whole differential coeflicient of {■i' — x){y'—y){^'—z) 

 with regard to the time D^V, and similarly for that of p, 



D, Y={x'-a){y'-y) (w' -'w) + {a;' -x) (z'-z) {■d-v) 



+ {2/-yW-z){u-u) 



Phil. Mag. S. 3. Vol. 30. No. 199. Feb. ISI-T. I 



