or 



where 



and 



Pien and Strom-Tejsen 



x-^ = -(p-r) tan y - 4> {p tan j3- r tan /3p) 

 - p{e' - 4>) tan /? - psd 



X - ^ = - [d + p (0' - 0) tan /3] , (25) 



d = (p- r) tan 7 + ps (0-0) + (p tan /3q- r tan /3p) , (26) 



R = [(x-^)2 + p2 + r2 - 2pr cos (0' -0)] 1/ ^ _ (27) 



When the propeller pitch varies with radius, the normal to the blade surface 

 will have a radial component corresponding to a force in the radial direction. 

 However, the radial component of the pressure dipole is ignored, since it is 

 usually very small and since a pressure dipole in a radial direction produces 

 very small downwash compared with a pressure dipole normal to the radial vec- 

 tor with the same strength. Hence cos i3q and sin /3p become the axial and tan- 

 gential components of the unit pressure dipole at Q{^,p,9). We break the tan- 

 gential component further into sin /3q cos {6' -0) and sin /Sq sin {9' - 4>) 

 corresponding to the tangential and radial directions at point P(x,r,0), respec- 

 tively. We denote these components by 1, m, and n as follows: 



1 = cos /3p , ^^ ,.,^j ^,^ .,^^^^, ^^ _ (28a) 



m = sin /3q cos X. , (28b) 



n = sin /3q sin k , (28c) 



where X = {6' - 4>). Likewise we have the three components of the distance vec- 

 tor from Q{^,p,e') to P(x,r,0) parallel to 1, m, and n respectively as follows: 



R^ = X - ^ = -(d+p\ tan /S) , (29a) 



R^ = p sin \ , (29b) 



R^ = r - p cos A. . (29c) 



By taking the gradient of both sides of Eq. (20) we obtain the acceleration com- 

 ponents Aj, A^, and Aj^ at point P(x,r,0)in the axial, tangential, and radial 

 directions respectively: 



A, = ^ - — + — ^ - (30a) 



98 



