46 APPENDIX. 



From these equations the diameters to the centre of pressure of the common wheel (given 

 in column 16 of the following table) have been calculated. 



In the new wheel the centre of pressure will be nearly in the centre of gravity, when the 

 paddle is totally immersed, the motion of the paddle being nearly vertical ; but in con- 

 sequence of the lower part coming sooner into and continuing longer in action, it must be 

 taken some distance below the centre of gravity. 



It is not easy to determine this by calculation ; but by a comparison of all circumstances 

 bearing upon this question, an allowance of one-eighth of the paddle has been made on this 

 account. 



It may be proper to observe, that in these wheels there is no relation between the diameter 

 of the polygon and the diameter to the centre of pressure, the paddles being differently hung 

 and differently shaped in the several vessels, particulars it has not been thought necessary 

 to .introduce into the table. 



The diameter to the centre of pressure, or effective diameter of the wheel, being known, we 

 at once deduce the excess of the velocity of the wheel over that of the vessel, or that at which 

 it recedes in the water to produce the resistance necessary for propelling the vessel. The rule 

 for ascertaining the amount of this resistance or pressure on the vertical paddle, is to multiply 

 the square of this velocity by the area of the paddle board and by G2| (the weight of a cubic 

 foot of water in Ibs.), and divide by 64J- ; the pressure upon a surface moving in a fluid being 

 equal to the weight of a column of water whose base is the area of the surface, and alti- 

 tude that through which a body must fall to acquire the velocity. This number, multiplied 

 by the velocity of the wheel, will express the power expended on the vertical paddles ; and 

 this divided by the whole power of the engine, will give the proportion consumed on the 

 vertical paddle given in column 18. 



In estimating the parts of the power of the engine exerted in any case, the number of 

 strokes made in a minute is compared with the actual number of strokes which ought to be 

 made for the engine to perform its whole duty. 



