210 THE DISPLACEMENT IN A CASE OF FLUID MOTION. 



then we can express \Vdt or x t in terms of elliptic functions of the first and 

 second kinds, 



win-re the position of the axis of the cylinder is expressed in terms of the 

 position of a molecule with respect to it. 



Now let us take a molecule originally on the axis of y, at a distance ij 

 from the origin, and let the cylinder begin to move from an infinite distance 

 on the negative side of the axis of x\ then 



$ = i}, and 2/J = ,/(4a f - , & 



and when the cylinder has passed from negative infinity to positive infinity in 

 the direction of x, then the coordinates of the molecule will be 



a(l c) 



-~ 



It appears from this expression, that after the passage of the cylinder e\vrv 

 particle is at the same distance as at first from the plane of xz, but that it 

 is carried forward in the direction of the motion of the cylinder by a quantity 

 which is infinite when y = 0, but finite for all other values of y. 



The motion of a particle at any instant is always inclined to the axis of 

 x at double the inclination of the line drawn to the axis of the cylinder. 

 Hence it is in the forward direction till the inclination of this line is 45", 

 backward from 45 to 135, and forward again afterwards. The forward motion 

 is slower than the backward motion, but lasts for a longer time, and it appears 

 that the final displacement of eveiy particle is in the forward direction. It 

 follows from this that the condition fulfilled by the fluid at an infinite distance 

 is not that of being contained in a fixed vessel ; for in that case there would 

 have been, on the whole, a displacement backwards equal to that of the cylinder 

 forwards. The problem actually solved differs from this only by the application 

 of an infinitely small forward velocity to the infinite mass of fluid such as 

 generate a finite momentum. 



In drawing the accompanying figures, I began by tracing the stream-lines 

 in Fig. 1, p. 211, by means of the intersections of a system of straight lines equi- 

 distant and parallel to the axis, with a system of circles touching the axis at 



