SECT, x.] OF STEAM NAVIGATION. 307 



none of the paddles would be in full action. A still more equable action will be 

 obtained by dividing the immersed arc into three ; beyond this I do not think the 

 advantage will be worth the extra expense, therefore I propose to give general 

 equations for any proportion, and particular rules for three to be immersed. 



637. To determine the radius of the wheel or the depth of the paddles, when 

 the number of the paddles is given, becomes an easy problem when the preceding 

 conditions are to be adhered to. For, put r = the radius B C, x = the depth 

 B D of the paddles, n their number, and a the number of parts into which the 



immersed arc is divided. Then ' - = the angle A C B, corresponding to half 

 the immersed arc, and 



rcos. i22 =CD, 

 



the cosine of the angle, being the depth from the centre of the wheel to the 

 surface of the water ; and, 



r cos. a } - = r- x; orr (l _ cos. ^ -)-=- BD, the depth of the paddles. 

 And, 



= r = B C, the radius of the wheel. 



T a LoO 



I cos. 



n 



From these equations we have the following rules for the case when three 

 paddles are immersed, or when a = 3. 



638. RULE i. To find the radius of the wheel, when the number and depth 

 of the paddles are given. Divide 540 by the number of paddles, which will give 

 the degrees in the angle contained by half the immersed arc. From unity sub- 

 tract the natural cosine of this angle, and the depth of the paddles divided by the 

 remainder will give the radius of the wheel. 



Or the radius of the wheel multiplied by the remainder will give the depth of 

 the paddles. 



639. RULE u. To find the number of paddles, when the radius of the wheel 

 and the depth of the paddles are given. Divide the depth of the paddles in feet, 

 by the radius of the wheel in feet, and subtract the quotient from unity. Find the 

 angle corresponding to the remainder as a natural cosine, and 540 divided by the 

 degrees in that angle is the number of paddles required. 



If the radius of the wheel be 8 feet, and the depth of the paddles 2 feet, then, 



1 - = -75 which is the cosine of the angle 41 4', 



o 



and rr* =13, the number of paddles. 



