124 APPENDIX. 



base is equal to the area of the surface, and height that due to the given velocity, multiplied 



into the cube of the natural sine of the angle of incidence. 



Let A B in the annexed figure be the horizontal projection of the given plane, which we 

 A. suppose to be vertical, and moving in the direction, and with the 

 velocity B C ; and draw C D perpendicular to A B. The pressure on 

 the given surface is by the former theorem proportional to the square 

 of its velocity through the fluid in a direction perpendicular to itself: 

 this velocity is here represented by the line C D, for the pressure is 



precisely the same as it would be if the surface moved in the direction of that line, and with a 



velocity equal to its length. Thus, calling v the velocity B C, i the angle of incidence ABC, 



t) ^ sin / ^ 



andjo the pressure on the surface perpendicular to its plane, p = a - , and the resistance 



2 9 



in the direction C B, which is equal to p sin. i, = a 



The power required to overcome this resistance with the velocity v, being equal to the 



77 " Sill t '* 



moment of the resistance or the pressure multiplied into the velocity, is a - v or 



y 



v ^ sin / '' 

 ^ , which is the same as if the surface moved perpendicularly to itself with the 



velocity v sin. i. This may therefore be called the effective velocity of the surface. 



These two propositions form the basis of all the calculations contained in the present 

 paper. 



CLASS I. PADDLE WHEELS WITH FIXED FLOATS. 



1. The Common Wheel. 



The construction of this wheel is so simple and so well known, that a description of it here 

 would be superfluous, having already referred to the diagram PL LXXIV. Fig. 1, which is 

 sufficient for our present purpose ; we shall therefore proceed at once to investigate the action 

 of the floats (which, for the sake of simplicity, we shall suppose to be perfectly radial) during 

 their motion through the water. 



Let R be the extreme radius of the wheel, r the radius to the inner edge of the floats, a the 

 height of the axis above the level of the water, < the angle which the surface of one of the 

 floats makes with the vertical at any given instant, x the distance of a given point .of the 

 surface from the axis, V its circumferential velocity, and v the velocity of the vessel, all the 

 measurements being given in feet, and the velocities in feet per second. The velocity of the 

 given point through the water in a direction perpendicular to the surface of the float, which we 

 have called its effective velocity, is equal to V v cos. $ ; for, in the figure page 121, A being 

 the given point, its effective velocity is A E ; and, the angle A B G being equal to A C D 

 which is equal to $ , and A B being equal to V and B G to v, we have 



AE=AB-BE =V-vcos. <f> . 



