192 Prof. Challis on the Hydrodynamical Theory of the 



I have obtained a general hydrodynamical equation in which the 

 factor enters as an unknown quantity. The present investiga- 

 tion does not require reference to that equation further than to 

 state that it serves to demonstrate the reality of the factor, and 

 consequently to establish the truth of the equation 



rdv dw\ fdw du\ /du dv\ , x , s 



which, as is known, is the general expression of the condition that 

 udx + vdy + wdz is integrable by a factor. 



27. I have recently learnt with some surprise from more than 

 one quarter that the equation (a), and, by consequence, the an- 

 tecedent views on which it is founded, are considered to be untrue 

 for reasons drawn from a discussion on certain hydrodynamical 

 questions which I had with Professor Stokes in the Philoso- 

 phical Magazine so long ago as 1842. Claiming to adopt views 

 expressed by Professor Stokes on that occasion, a correspondent 

 sends me the following argument relative to the equation (a). 

 Conceive to be impressed on all parts of the fluid the arbitrary 

 constant velocities u, jB. y in the directions of the axes of coor- 

 dinates. Then the equation becomes 



, ./dv dw\ , n\/d w du\ , .(du dv\ _ 



which, since a, (3, 7 are perfectly arbitrary, cannot be true unless 



dv dw _ dw du _ du dv _ 

 dz dy ' dx dz ' ' dy dx~ ' 



that is, unless in every instance of the motion of a fluid 

 udx + vdy + wdz is an exact differential. As this is certainly not 

 the case, it is concluded that the equation (a) is untrue. 



28. The answer to this argument is that the equation (a) was 

 deduced on the principle of its being exclusively applicable to mo- 

 tions which are peculiar to a fluid, and which, consequently, a 

 solid is not capable of, the motions, namely, by which the parts 

 of a fluid mass in motion can change their relative positions. 

 This is the sole raison d'etre of the equation. Hence the intro- 

 duction of the velocities a, /3, y common to all the parts of the 

 fluid is a violation of the principle on which it is founded ; or 

 rather the above argument is a proof a posteriori that the equa- 

 tion excludes such common velocities. If, therefore, that equation 

 be satisfied, there is no need to " define " the velocity that may 

 be common to all the parts of the fluid; for either such motion 

 takes place under given conditions, and is consequently known, 

 or, if not known and not knowable (whether it be due to the 

 earth's rotatory and orbital motions, or to the motion of the 



