Mass carried forward by a Vortex. 607 



In connexion with the construction circles k = const., they 

 cut the axis of x at points 



i-k ' i+y 

 x - a i + k' > a i-k" 



so that 



2k' . 1 + k' 2 2 



rad= ~ , distance of centre from axis = — ^ — = p — !• 



Toroidal configuration. — Passing now to the consideration 

 of the case where the second region is ring-shaped, the 

 boundary is given by the loop of that stream line which cuts 

 itself in the equatorial plane, at a point where the velocity is 

 zero. The value of the stream function for this is clearly 

 negative, since in the aperture between the centre and this 

 point the velocity is everywhere negative. As before the- 

 stream function is given by 



Under given conditions, z. e. V given, we have to find the 

 value of x which makes dyJdx—Q when j/ = 0. Substituting 

 this value of x and ?/ = in % then gives the constant value 

 (say ^) of the bounding stream line, and the equation of the 

 latter is % = %i- The finding of the root of d%/dx can be 

 carried out by ordinary approximation when numerical ex- 

 amples are required. Our present purpose, however, is not 

 so much to get the result for a given state (value of b/a) as to 

 follow the changes as the states vary. For this purpose it is 

 best to choose positions for sets of nodes, and calculate the 

 values of U required to satisfy d%/dx = 0. The corresponding 

 value of b/a is then found from U. For instance choose k,. 

 where x/a—(l—k')j(l + k f ) since now x<a. 

 Then 



1 /a v // \ d3L dk Tr A 



dk kk' 



~r = tt when t/ = 0. 

 dx 2x 



Thus X + k k' ^ - 2aY (§* = 0. 



It follows after an easy reduction that 



4 A' 



