Prof. Challis on the Principles of Hydrodynamics. 241 



The " first argument " is merely a particular instance of the 

 second^ and docs not require to be considered separately. 



The following is the " second argument." If the axiom that 

 "the directions of motion in each element of the fluid mass may 

 at all times be cut at right angles by a continuous surface/' be 

 admitted to be of general application, it mil follow that the 

 quantity udw + vdij + wdz is always integrable, either of itself or 

 by a factor. But on combining motion for which that quantity 

 is integrable only by a factor with a motion of translation of the 

 whole mass, so that the components of the resulting motion are 

 u + a, v + b,w + c, Prof. Stokes proves that {u + a)dx-\-{v-'rb)dy 

 + {w + c)d3 is not integrable by a factor, and thus appears to 

 controvei-t the position that the axiom is of general application. 

 To this reasoning an answer may readily be given. The result 

 is arrived at by introducing a motion of translation of the whole 

 mass, which is precisely the kind of motion which it is mmeces- 

 sary to include in the hydrodynamical equations. The motions 

 we are concerned with in hydrodynamics alter in some way the 

 pressure, densitj-, or mutual distances of the fluid particles ; but 

 a motion of translation has no such effect, and does not possess 

 a single characteristic in common with the motion proper to a 

 fluid. In the expression above, the terms udx + vdy + wdz have 

 no relation whatever to the terms adx + bdy + cdz ; and it is in 

 perfect accordance with this circumstance (which Professor Stokes 

 appears to have entirely overlooked), that the sum of the two sets 

 of terms does not admit of being made integrable by a factor. 

 On account of the motion which alters the density, pressure, or 

 mutual distances of the fluid particles being wholly independent 

 of any motion of translation of the whole mass, it is a prelimi- 

 naiy step in any hydrodynamical problem to get rid of the latter, 

 which may always be done either by conceiving the origin of 

 coordinates to partake of the motion of translation, or by im- 

 pressing it on the mass in a direction opposite to that in which 

 it takes place. The residual motion, or the dificrence between 

 the motions of the particles and the motion of the origin of co- 

 ordinates, is all that is considered in hydi-odynamics ; and if the 

 axiom under discussion applies to this motion, it possesses the 

 requisite degree of generality. Its not applying to the combi- 

 nati(m of a motion of translation with the motions proper to 

 hydrodynamics, is only a ])roof of its strictly hydrodynamical 

 character. For these reasons I cannot admit that there is any 

 force in Professor Stokes's argument, nor can I attach any im- 

 portance to the remark by which it is followed. 



Cambridf^e Obsen-atory, 

 February 4, 18.51. 



Phil. Mag. S. 4, Vol. 1. No. 3. March 185 1 . II 



