Prof. Challis on the Principles of Hydrodynamics. 441 



and then eliminating by a known process p, u, v, w from the three 

 equations (a), [jS), (7), the result will be^ 



Xdx^ dx^ dx dy dxdy dy^ dy^) 

 _ „^ ^ _/B #! ^ 



^ ■ (S) 



dz dzdt •' 'dz'' dz"^ 



But since, from what has already been said, this equation applies 

 only to points on the axis of z, or immediately contiguous to it, 



the terms involving -j-^ and -j- will be infinitely less than the 



other terms. And again, as the value /=1 results from the 



values a?=0, y=0, which make -^ =0, and -j- =0, it follows 



that that value of / is either a maximum or a minimum. The 

 supposition of a minimum leads to contradictoiy results, and is 

 by that circumstance excluded from the investigation. Hence, 

 supposing that 



dj d^ m 



dx' "^ </y2 - g2> 



and omitting in (8) the terms involving -j^ and -j-, we have for 

 the motion along the axis. 



The arbitrary quantity r'(/) disappears if F(^) be supposed to be 

 zero or a constant, and this supposition is required by the motion 

 which is the subject of this investigation, which is independent 

 of any arbitrary circumstance. Omitting, therefore, F'(/), the 

 equation just obtained possesses the remarkable property of being 

 satisfied by motion along the axis, such that the density and 

 velocity existing at any instant at any point arc propagated with- 

 out alteration at a certain uniform rate. This property I pro- 

 ceed to demonstrate. 



On the supposition that the motion is of the kind above 

 described, the density {p) must satisfy the equation 



