THE DISPLACEMENT IN A CASE OF FLUID MOTION. 209 



of the cylinder ; then it is easy to shew that, if x is the value of x for the 

 axis of the cylinder, and x that of the point, and 



<h = -(x-x) and d-ll-- 



V(f> will satisfy the conditions of the velocity-potential, and V-fy that of the 

 stream function* ; and, since the expression for \|> does not contain the time, 

 its value will remain constant for a molecule during the whole of its motion. 



If we consider the position of a particle as determined by the values of 

 z, r, and t/f, then z and $ will remain constant during the motion, and we have 

 only to find r in terms of the time. For this purpose we observe that, if we 

 put i| in polar coordinates, it becomes 



= ( 1 2 ) r sin 6, 



dr V d^ I a*\ 



and -y = - T = V 1 --^ cos 0. 



dt r d6 \ ry 



Expressing cos 6 in terms of r and i/>, this becomes 



If we make v /(4a 2 + 1/ 2 ) + i/r = 2/3, and |- 2 = c, 



then /8 will be the value of y when the axis of the cylinder is abreast of the 

 particle, and 



and if we now use instead of r a new anular variable sucn 





s _ 



A r 



f The velocity-potential is a quantity such that its rate of variation along any line is equal to 

 the velocity of the fluid resolved in the same direction. Whenever the motion of the fluid is irro- 

 tational, there is a velocity-potential. 



The stream function exists in every case of the motion of an incompressible fluid in two dimen- 

 sions, and is such that the total instantaneous flow across any curve, referred to unit of time, is 

 equal to the difference of the values of the stream function at the extremities of the curve. 



VOL. II. 27 





