A General Theory for Marine Propellers 



If the difference between /S^ and /3 is neglected, a, b, and c in Eqs. (73) 

 are independent of 6 - 4>, and these integrations can be carried out functionally. 

 Since recurrence formulas are available to express J3'" in terms of J™"' and 

 J^"^ for m > 2, J3" can be obtained rapidly after Jj", 53^, and :i^ have been com- 

 puted. Likewise it is necessary to obtain only a few terms of 1^ or J^^g by in- 

 tegration. The remaining can be obtained by recurrence formulas. 



If the difference between /Sq and /3 is taken into account, a is independent 

 of - (p, but b and c are functions of 6^ - '^. However, the expression of the 

 distance factor R in Ki^, K^^, and Knj can still be expressed as the square root 

 of a second degree polynomial of - 4>. After substituting - 4> for y we have 

 from Eq. (42) 



R = [a(^-0)2 + 2h{0-4>) + c]^^^ . (74) 



From Eq. (26) we write 



d = do + d^{0-^) , (75) 



where 



dp = (p- r) tan 7 + (p tan /3q - r tan /5p) (76a) 



dj = ps = p (tan /3q- tan ^6) . (76b) 



From Eqs. (43) and (75) ' • ' • 



■ ■■ ^". a = p2 tan^ /3 - 2pTC^ , ,. "-','" .".,'' ' ('77a) 



b = bp + bi(5-0) , , , ■ '■!,.: '",_ ;...■,;■ (77b) 



_. ..'y c = e^ + ei(0-0) + 62(0-0)2 , ;;-,.,., ., ■ (77c) 



where '■ "•' ' ' : ' •. ' ^ - ; :■.■■■:.' . . :'-.. 



bg = p^ka tan^ /3 - 2prCj 4 d^p tan /3 , ■• ' = .•- 



b^ = dj p tan /S , • 



Bq = p2 + r^ - 2prcQ + (pka tan /S)^ + d^^ + 2dQpka tan /3 , 

 e^ = dgdj + djpka tan /3 , 

 e^ = d,' . 

 Substituting the previous expressions of a, b, and c into Eq. (74) we obtain 

 R = [a'(«9-(/.)2 + 2h'(0-4>) + c']*''^ , 

 111 



