Prof. Challis on the Principles of Hydrodynamics. 35 



do' 

 To integrate these equations, let m= ~, Substituting this 



value in (4.) and integrating, we have 



a'Nap.logp+f +^.^^'=FW. 



Or, if q = q'—J F{t)dt, and consequently m= -^, the equation is 



Hence, eliminating p from (5.) by this last equation, we obtain 



dxA dxV dx dxdt df" ' 



which equation is not generally integrable, but is satisfied by the 

 following particular integral, 



u=f{x — a-\-u.t). 



Now although this integral seems to have been arrived at by an 

 unexceptionable course of reasoning, and might be expected to 

 admit of interpretation consistent with fluid motion, yet upon 

 trial this is not found to be the case. Suppose, for instance, that 



• 27r, ^, 



M = OTsni -TT- [x — a + M.f). 



Then if x = fl^+ — , w = 0: and if a7=(a + m)<+ :r, u = m. But 

 these two values of x are the same if 



that is, H t— -r-. 

 4m 



Hence, at the same distance from the origin the velocity may 

 be zero, and may have its maximum value at the same moment. 

 This result is a contradiction per se, having no reference to any 

 physical circumstance, but indicating that a false stej) has been 

 taken in the i-easoning. It will be seen hereafter that this con- 

 tradiction has an important bearing on the analytical theoiy of 

 fluid motion. At present I propose to adduce other examples 

 of like failures, for the purpose of showing distinctly that the 

 princii)le of the foregoing reasoning is somewhere at fault. 



Example 11. Let, as in the preceding example, /? = a^p, and 

 let the velocity and density be functions of the distance from a 



D2 



