Pien and Strom-Tejsen 

 where 



a'=a+2bj+e2, 

 b' = bo + ei/2 , 

 c' = e^ . 



Also 



ac - b2 = a[eo + 6^(^-0) + 63 ( - 0)^ ] - [b^ + b^ ( - 0)]2 



. f, + f^(6?-0) + f2(^-0)' , (78) 



where 



fj = ae^ - 2bjbQ , 



^2 = aej - bj^ . ^^ 



With these expressions it is clear that the chordwise integration over the 

 blade can be carried out easily with I3" or l^"" as the integrand, except for 13°, 

 13^ and 15°, where the factor ac - b^ is also involved in the denominator of the 

 integrand. It is possible to obtain functional solutions even for these special 

 cases. However the functional solution is so complicated in each case that it is 

 easier to carry out the integration numerically. 



The last integration in the radial direction with respect to p is carried out 

 numerically. A computer program is in preparation for computing at any time 

 the induced velocity components at point P(x,r,0) from the action of a moving 

 blade. The total induced velocity due to the action of a propeller is, of course, 

 the sum of the contributions from all the blades. This program can be used 

 either for propeller design or for propeller performance prediction or simply 

 for computing the induced velocity in the field. 



APPLICATION OF THE THEORY 



Propeller Design Problem 



In a design problem, the blade contour is chosen from consideration of 

 cavitation and blade strength. The path of the propeller is known from the pro- 

 peller forward speed and the angular velocity. The lift distribution depends 

 upon the specified thrust distribution over the blade and the orientation of the 

 blade in space. With the blade contour orientation which defines the blade pitch 

 distribution not known, the pressure dipole distribution is not known. Therefore, 

 an iterative procedure is necessary. To start with, the advance angle /3 at each 



112 



