A General Theory for Marine Propellers 



where Cg = cos ka, Cj = a^ cos ka - sin ka, and C2 = a^ cos ka - a^ sin ka. 

 Equations (28) become 



1 = cos /3q , (38a) 



m = Cq sin /3q + Cj sin /Sq y + c^ sin /3>q y^ , (38b) 



n = Sg sin /3q + s j sin /3q y + s^ sin /3q y^ . ' (38c) 



Equations (29) become 



Rj = -(d + pX. tan /3) = -d - pka tan fi - p tan /3 y 



= Xg + x^y, with Xg = - ( d + pka tan /3), Xj = -p tan /3 , (39a) 



^m = -^^0 + P^iy + PSjy^ (39b) 



^n = ^ " '^'^o " ^^ly ~ ^^2y^ • =■ ■, ' (39c) 



From Eqs, (31), (37b), and (38) 



'■ M - Mg + Mjy + Mjy^ , _ (40) 



where 



Mg = Xg COS /3q + rsg sin /3q , (41a) 



Mj = rsj sin /5q + Xj cos /3q , (41b) 



M2 = rsj sin /3q . ._ . (41c) 



Equation (27) becomes 



R = (ay2 + 2by+ c)i/2 _ (42) 



where ^ .• ;, . , 



a = p2 tan^ /3 - 2prc2 , ■ V (43a) 



b = p tan /3 (d + pka tan /S) - 2prCj , (43b) 



c = (d + pka tan /3)2 + p2 ^ r^ - 2prCg . - (43c) 



Now within each interval, say from ka to (k+ l) a, we define Kj^ as 



101 



