132 134] VELOCITY-POTENTIAL DUE TO A VORTEX. 157 



where 



Here I, m, n denote the normal to the element dS of a surface 

 bounded by the vortex-filament. 



The equation (22) may be otherwise written 



(28) - 



where ^ denotes the angle between r and the normal I, m, n. 

 Since - j -- is the elementary solid angle subtended by dS at 

 (x, y, z\ we see that the velocity-potential at any point due to 



w a 



a single re-entrant vortex is equal to the product of into 



the solid angle which any surface bounded by the vortex subtends 

 at that point. 



Since this solid angle changes by 4?r when the point in 

 question describes a circuit embracing the vortex, the value of $ 

 given by (23) is cyclic, the cyclic constant being twice the strength 

 of the vortex. Compare Art. 127. 



Dynamical Theorems. 



134. In the theorems which follow, we assume that the 

 external impressed forces have a single-valued potential T 7 , and 

 that p is either a constant or a function of p only. 



We first consider any terminated line AB drawn in the fluid, 

 and suppose every point of this line to move with the velocity 

 of the fluid at that point. In other words the line moves so as 

 to consist always of the same chain of particles. We proceed 

 to calculate the rate at which the flow along this line, from A 

 to B, is increasing. If dx, dy, dz be the projections on the axes 

 of co-ordinates of an element of the line, we have, with our 

 previous notation, 



3 , N du , ddx 

 ( M &amp;lt;fe.) = _&amp;lt;fo +M _. 



