SECT. X.] 



OF STEAM NAVIGATION. 



299 



Let B A C = a be the angle which the screw forms with a line A B perpen- 

 dicular to its axis ; then during the time the boat would move from C to B, a point 



FIG. 26. 



in the surface must move from B to A, otherwise it would retard the boat ; and, in 

 order that it may be effective, it must move at some greater velocity. But the 

 velocity of the boat, v, is to that of a point in the surface when no effect is pro- 

 duced, as B C : A B : : v : n A = - - Hence the actual effective velocity 



tan. a 



must be, 



V - 



tan. a 



V tan, a v 

 tan. a 



2 IT x 



Let x be the variable radius of the cylinder, then - - = the length of the 



spiral, and '^- = the differential of its area. Its resistance is therefore, 



v (V tan, a >)* (2 sin, a 3 + sin. aPjxdx 



cos. a tan. * a 



when the vessel is at rest ; and when it is in motion, it increases in the ratio of 

 : v ; hence, 



TT v (V tan. a v) (2 sin. 2 a + sin. a) x d x = the differential of 



V tan. a v 

 tan. a 



the resistance. 



The integral gives, 



|- IT v x 3 (V tan. a v) (2 sin. 2 a + sin. a) the resistance. 

 This resistance is to the effect to impel the boat, as the radius is to tan. a ; 

 hence, 



^ IT x 3 v (V tan. a vf (2 sin. a" + sin. a) tan. a = the force, 

 and % TT x z v z (V tan. a -- v) (2 sin. 2 a + sin. a) tan. a = the effect, 

 which should be equal to the resistance of the vessel. 



It is a maximum when v s (V tan. a v) = a max. : that is, when V = ~ - 



2 tan. a 



Hence the effect at the maximum is ^ w x~ v 3 (2 sin. 2 a + sin. a) tan. a. 



i ., , , 3 ir x- v 3 (2 sin. 2 a + sin. a) , . , 



But the power to produce it must be, - g - , its velocity being 



