J^^^t.' Furmernis^ Principles of Hydrodynamics, 



""' in Reply to Professor Stokes. By Professor Challis*. 



AGAINST the general proposition that udx + vdy + wdz is 

 integrable by a factor. Professor Stokes broiight the ob- 

 jection that the proposition is not true if u, v, rv include a motion 

 of translation of the whole fluid mass. I replied (in the March 

 Number) that the objection has no force, because such a motion 

 of translation may be, and usually is, left out of consideration in 

 hydrodynamics. The motion of translation of the solar system, 

 for instance, is not taken into account, not being accurately 

 known either in magnitude or direction ; and as this motion of 

 translation is of necessity eliminated, so we may eliminate a 

 motion of translation m any case. Professor Stokes is under a 

 misapprehension in supposing that I have asserted the elimina- 

 tion of the motion of translation to be necessarily a preliminary 

 step. I made no such assertion. When that motion takes place 

 under known circumstances, it may be included in the values of 

 u, V, w. But at the same time that this is done, the principle 

 must be recognized, that the motion of translation of the whole 

 mass, which the mass might receive if it were solid, and the 

 residual motion, which exclusively belongs to it as a fluid, are 

 independent of each other. If this principle be attended to, the 

 terms relating to the motion of translation separate themselves 

 by the course of the reasoning from those relating to the remain- 

 ing motion, and the result is the same as if the motion of trans- 

 lation were in the first instance eliminated. 



The new argument brought forward by Professor Stokes in 

 the May Number is little else than a reproduction of the former 

 argument in another form. The instance of motion adduced is 

 that of a mass of fluid revolving with a given angular motion 

 about a fixed axis. According to the principle of the separability 

 of the parts of a fluid without assignable force, the fluid in this 

 very particular case of motion may be conceived to be separated 

 by indefinitely thin cylindrical partitions into an unlimited num- 

 ber of cylindrical shells having for conunon axis the axis of rota- 

 tion. An arbitrary motion of translation parallel to the axis may 

 be given to any one of these shells, without affecting its motion 

 of rotation, and quite independently of any arbitrary motions of 

 translation parallel to the same axis, which may be given to the 

 contiguous shells. Hence for this instance udx + i^dy + ivdz can- 

 not be made integrable by a factor, because the quantity lu, 

 which expresses the motion of translation of any one of the 

 cylindrical shells, is altogether an arbitrary quantity, and is not 



* CommuBicated by the Author. 



