194 Prof. Challis on the Hydrodynamical Theory of the 



pressed by the equation 



du dv dw _ 

 doc dy dz ' 



which is true for a compressible fluid inclusively of small terms 

 of the second order ; so that the subsequent reasoning, although 

 strictly applicable to an incompressible fluid, may be taken to 

 apply to the sether. Now, from the known expressions for 



-~> ~~y ~- for steady motions of an incompressible fluid, it 

 dx dy dz J r 



will readily be found that 



\_t± _ \J\ )J _ thg centrifugal force. 



dr r ° 



31. These results are independent of the forms of the func- 

 tions f(r) and F(r) and of any relation between them. But 

 since the assumed values of u, v, w do not make udx + vdy + wdz 

 an exact differential, according to the principles maintained above, 

 they must be such as to satisfy the equation [a). By substitu- 

 ting them in that equation, and integrating, the result is 



/W _e 



F(r) r 

 c being the arbitrary constant introduced by the integration. 

 We have thus demonstrated that the current must be such as to 

 satisfy the relation between the velocities/(r) and F(r) indicated 

 by this equation. 



32. By taking account of this relation the equation 



udx + vdy + wdz = 

 gives 



f(r) c 



dz = ~ ppjr) (ydoo-ocdy) = -j {xdy-ydx). 



Hence, by integration, 



z = ctan~ 1 — -\-b. 



x 



which is the general equation of the surfaces of displacement, 



the orthogonal trajectories of which determine the directions of 



v 

 the motion. If tan -1 - =0, and r x be a given distance from the 



x 



axis, we have 



r \Q _ r \ 

 z~^b~c ; 



which shows that the motion in the cylindrical surface of radius 

 r, consists of spiral motions the directions of which make with 



