Rectilinear Fluid Motion^ in Reply to Mr. Stokes. 425 



+ wdz = 0, then by a common step in analytical reasoning, 



du . dv , dw j _ .j , , , , , 



d~t ^ + It y + dT } provided dx 9 dy, dz do not 



vary with the time. Hence as it is proved above that dx, dy, 

 dz do not vary with the time in the equation udx + vdy 

 -f tods =0, when the left-hand side is an exact differential 



(d <p), it appears that d $ = 0, and d . —r- = 0, are differ- 



£ 



ential equations of the same curve surface. The following is 

 an instance. Let the velocity V be directed to or from a 

 fixed centre whose coordinates are a, /3, y, and be the same 

 at the same distance (r) from the centre at a given time. 

 Then because 



tidx+vdy+wdz, orV.f dx + ~ — ? dy-\ ^dzJ^O, 



it follows that 



du . dv 7 , dw j 



-dT dx+ -n dy+ -dt d *> or 



dx + 

 dt \ r r 



^+ -f 1 dz ) =°» 



and these are differential equations of the same curve surface. 



3. In answer to the third argument it is sufficient to say, 

 that any proposition proved respecting Jluid motion, that is, 

 motion by which the parts of the fluid alter their relative po- 

 sitions, cannot be affected by motion which is common to all 

 the parts. There is no dependence of the one kind of motion 

 on the other. The equation of continuity and the equation 

 derived from D'Alembert's principle are identically satisfied 

 by the latter kind of motion, which must be considered to be 

 eliminated before any use is made of those equations for de- 

 termining fluid motion. 



4. The solution here given of a bydrodynamical problem 

 is inadmissible on this ground. If a direct solution of the 

 problem had been attempted, it would have been found ne- 

 cessary to inquire whether ud w + v dy + wd z were an ex- 

 act differential for that instance ; and no mode of solution 

 could evade the consideration of this question, unless the fluid 

 were supposed to be confined between two cylindrical surfaces 

 indefinitely near each other, and having hyperbolic bases. 

 As in Mr. Stokes's solution that question is. not considered, 

 I conclude that it only applies to the limited case. 



There is another point connected with this subject, and of 

 no little consequence in the mathematical theory of fluid mo- 



