126 APPENDIX. 



/3 being the angle at which the upper edge of the float enters and leaves the water. The 

 difference multiplied by 2 m, the number of floats in the two wheels, will express the power 

 expended upon them in ffis. raised one foot per second, which is reduced to horse powers by 

 multiplying by 60, the number of seconds in a minute., and dividing by 33000. We find thus 

 for the general expression of the power of the engines of a steam vessel transmitted to the 

 paddle wheels, 



. . (1) 



To find the portion of this power which is effective in propelling the vessel, we must first 

 find the mean horizontal pressure on the floats, which, multiplied into the velocity of the 

 vessel, will be equal to the effective power required. 



The horizontal pressure on one of the floats in any given position is equal to the product 

 of the total pressure by the cosine of its inclination, or 



R 

 jr'ri'bw 



900 x 2ff 



I [xp cos. $] 2 cos. $ dx. 

 \J 



COS. <p 



This is equal to 



TJ- + a?p a p 2 cos. 



the mean value of which during an entire revolution is, 



2700x20,7 V cos.<f> 







Integrating and simplifying, we find the mean horizontal pressure on one of the floats, 

 supposed to extend to the surface of the water, to be equal to 



7TW 2 bw ( i \ I \1 



,,-.. r { 6a?p3R'p3ap i ) .+ (2R 3 + 4Rp 2 3 Rap 2Ra? ap 2 cos.a) sin. >. 

 5400 x2y l\ / v / J 



The quantity to be deducted in cases where the float is immersed above its upper edge, 

 is found in the same manner as it was done for the total power, and the remainder, which 

 then expresses in Ibs. the mean horizontal pressure on one of the floats, multiplied into the 

 velocity of the vessel, which is equal to 2 ir p n, in feet per minute, will give the propelling 

 effect due to the float in tbs. raised one foot per minute. We have thus, 



sn. a 



sin. /3 



Multiplying by 2 m, and dividing by 33000, we find, for the general expression of the effective 

 power, or that portion of the power of the engines which is employed in propelling the 

 vessel, 



