10 Rev. J. Challis on the Motion of Fluids. 



a function of r and t, is p = ^- -f /(), and the velocity 

 From what has been said, these ex- 



or r 



pressions for <p and co will apply to any point of a mass of 

 fluid, moving so that d 4> is an exact differential. But in ge- 

 neral F (t) and f (t) can be considered constant at a given 

 time only for values of r restricted within limits indefinitely 

 near each other. Let r' be a value indefinitely near to r. 



Then*'-? = F (t) ( - - -1)= -*&.p- f ) = (,<-r). 



Here ? J r may be considered the increment ds of a line s 9 

 drawn continually in the direction of the motions of the par- 

 ticles through which it passes. Hence d$ = ads; and 

 p =fa> ds -f x (0> tne i nte gral being taken in regard to an 

 arbitrary portion of the line 5. The two expressions for <$> 

 thus obtained, have a relation to each other, analogous to that 

 between the two expressions which the general and the par- 

 ticular solutions of a differential equation of the first order 

 give for the same variable. By equating these values of f, 



But in the example before us, if y = the distance of any 

 point from the axis, r, being the length of the portion of a 

 tangent to s intercepted between the point and the axis, will 



be y --. Therefore, 



or ' , 'V = ./* + *co 



Hence - r =fwds + x (0 ~/(0 5 and .'. o = 



y y **o > 



Now the particles in contact with the surface of the cone 

 must move in straight lines directed to its vertex : and if 2a 



= its vertical angle, - = sin . Hence to = ^-^ that 



j' 



is, the velocity varies inversely as the square of j/, and conse- 

 quently inversely as the square of the distance from the vertex. 

 Therefore if we conceive a conical surface to have the same 

 vertex and axis as that which contains the fluid, and to have 

 a vertical angle, less by an indefinitely small angle than Vot, 

 the fluid contiguous to the containing surface will move as if 



included 



