212 THE DISPLACEMENT IN A CASE OF FLUID MOTION. 



the origin and having their radii as the reciprocals of the natural numbers. 

 (See Prof. Rankine's Papers on Stream-Lines in the Phil. Ti-ans.) 



The cylinder is f inch radius, and the stream-lines are originally ^ inch 

 apart. 



I then calculated the coordinates, x and y, of the final form of a transverse 

 straight line from the values of the complete elliptic functions for values of c 

 corresponding to every 5*. The result is given in the continuous curve on the 

 left of Fig. 2, p. 213. 



I then traced the path of a particle in contact with the cylinder from the 

 equation 



where x=x 9 + acos& and y = ctBm0. 



The form of the path is the curve nearest the axis in Fig. 3. The dots 

 indicate the positions at equal intervals of time. 



The paths of particles not in contact with the cylinder might be calculated 

 from Legendre's tables for incomplete functions, which I have not got. 



I have therefore drawn them by eye so as to fulfil the following con- 

 ditions : 



The radius of curvature is -5. ., fi > which, when y is large compared 



ct sni. (7 ~T* y 



with a, becomes nearly . 



2y 



The paths of particles at a great distance from the axis are therefore very 

 nearly circles. 



To draw the paths of intermediate particles, I observed that their two 

 extremities must lie at the same distance from the axis of x as the asymptote 

 of a certain stream-line, and the middle point of the path at a distance equal 

 to that of the same stream-line when abreast of the cylinder ; and, finally, that 

 the distance between the extremities is the same as that given in Fig. 2. 



In this way I drew the paths of different particles in Fig. 3. I then 

 transferred these to Fig. 2, to shew the paths of a series of particles, originally 

 in a straight line, and finally in the curve already described. 



