Bavin, Vashkevich, and Miniovich 



is the distance between the field point P(Xq, r^,-/^) and the point of helical surface 

 Q whose coordinates are ^ = Vp(t -r), r', and j>=aj(t-T); t is the time during 

 which the lifting line has moved from point Q to its position at time t (Fig. 1); 

 and n is the normal direction at point Q. 



The magnitude of the potential discontinuity [^l] at point Q equals the cir- 

 culation around the circuit I embracing the part of the helical surface between 

 this point and the bound vortex (Fig. 2). If the surface on which the circuit l 

 lies is taken to cross the helicoidal surface along a helical line passing through 

 the point Q, then the circulation in the circuit I will be equal to the sum of the 

 bound vortex strength r(r ' , t) at time t and the total strength of all the free 

 vortices distributed between the bound vortex and the point Q, i.e., radial free 

 vortices shed by the blade during the time r. But according to Thomson's theo- 

 rem the aforementioned sum is equal to the strength of the bound vortex at time 

 (t-T) when the latter was at the point Q. Then [i>^] =r(r',t-T) and 



tV f r r(r',t-r)D(x^,7„)R-i drdr' , (1) 



where (1) 



D(Xn.7j --[^r -- ^ - 



In terms of the linearized theory the pressure at an arbitrary point of the field 

 (exclusive of any hydrostatic increments) is 



where p is the density of the fluid. Then 



(2) 



R Q 00 

 Rp CO 



I I |7[r(r',t-r)D(x„,7„)R-i]dTdr' 



P 

 Arr 



Ro -a)r (Xq-V t) + V r^ sin (7^ - «t ) 

 — rrr'.t) dr' 



T=0 



Introduction of the coordinates x = x^ - Vpt , r = r^, and 7 = Tq yields 



Rq -cor'x - VpF sin (7- 6) 



p = ^ r{T' ,6) dr' , (3) 



4^ Jq [x2 + r2 + r'2 - 2rr' cos {y-d)]^^^ 



