and the right side of the matrix is 



B P = JT C * V 



where C„ represents the right side of simultaneous Equations (36). The solution of 

 the simultaneous equation 



yi.. b ,. .. +y~b b ,. . + yi. b ... = b os) 



X, ij Pq(ij) j/^ ^m pq(£m) / , k pq(k) p 

 P = 1, 2,.... 



is the desired least squares solution for the cavity source. 



IMAGES FOR THE HUB 

 To satisfy the hub boundary condition 



u = on r = r TT 

 r H 



the two image systems for vortex and source distributions were considered separately. 

 The hub images for the vortex distribution are the same as those Kerwin used. 



That is, the same vortex strength as that distributed on the blade (x,r ,9) is used 



2 

 at the image point (x ,r„,9) where r • r„ = r . Then the trailing vortices tend to 

 1 z 1 z n 



cancel the radial velocity on the hub. This is an approximation to a two-dimensional 



vortex outside a circular cylinder. 



The hub images for the source distribution need special consideration because 



the cavity sources for a supercavitating propeller are much stronger than those of 



subcavitating propellers. The first approximation of the image source may be taken 



from an approximation of the hub by a sphere. The image system for a point source 



2 

 of a unit strength at r = r of a sphere is a point source at r = r /r with the 



1 Z n 1 



strength r „/r and a line sink stretching from r = r„ to the center of the sphere 



ti 1 Z 



with the strength 1/r . In this way the total absolute strength of the line sink is 

 H 



equal to the point source at r = r . Thus, on the sphere, r = r , the normal 



^ H 



16 



