154 APPENDIX. 



the vertical diameter at the point S', and let 8 = the angle S S' S" contained between 

 the surface of the float and the vertical. Let S E, perpendicular to C S = the circum- 

 ferential velocity of the spindle = V, and the horizontal line S G = the velocity of the 

 vessel = v. Through the point S draw a straight line F H perpendicular to A B, and 

 from the points E and G draw E F and G H perpendicular to F H, and meeting it at 

 the points F and H ; the difference between F S and S H will be the effective velocity of 

 the spindle. Now, the angle E S F being equal to the angle S S' S" or 8, we have F S = 

 V cos. 8, and, because the angle G S H = 8, we have also S H = v cos. 8 ; therefore, the 

 effective velocity of the spindle = ( V v) cos. 8, which is always positive, for the speed 

 of the vessel is always less than the circumferential velocity of the spindle. Through 

 the points S' and F draw a straight line S' K, and from any point A of the line A B draw 

 another straight line A K parallel to S F, and intersecting the former at the point K ; 

 A K S H will be the effective velocity of the point A : for, since the float always radiates 

 from S', we may consider that point as the centre of revolution, in which case S F would 

 represent the circumferential velocity of the spindle, and consequently A K that of the point 

 A. Now, if we combine this with the horizontal velocity S G, and resolve the resultant into 

 two others, one perpendicular and the other parallel to A B, the former, which is the effective 

 velocity of the given point, will be found to be equal to A K S H. Let z = the distance 

 of the given point from the spindle, taken positively towards the lower edge of the float, and 

 negatively towards its upper edge ; then we shall have 



AK : SF :: AS' : S S', 

 or, 



A K : V cos. 8 : : 2 R cos. 8 + z : 2 R cos. 8, 

 whence, 



A K = JL [2 R cos. 8 + z] , 

 and, deducting S H, we shall find the effective velocity to be 



- [2 R cos. 8 + z] v cos. 8, 

 2 H 



or, putting for V and v their values, 



Effective velocity = 2L^L [2 (R - p) cos. * + z] . 

 60 



From this equation it appears that the difference between the effective velocities of any two 

 points on the surface of the float is constant during the whole of the stroke, and that the 

 effective velocity of the spindle increases in the same ratio as cos. 8, therefore the effective 

 velocity of any point, and consequently the pressure upon it, increases from the moment it enters 

 the water until it arrives at the middle of the stroke, and then decreases again until it leaves 

 the water. This property causes the shock to be much less than with the common wheel, of 

 which the floats experience the greatest resistance at the beginning and end of the stroke. 



A curious property of this wheel is that the effective resistance is always equal to the tan- 

 gential resistance at the spindle, which is the resistance to be overcome by the engines ; so 

 that the propelling effect is to the power employed as the radius of the rolling circle, or p, is to 

 the radius of the wheel R, which ratio has not been attained by any of the foregoing varieties 

 of wheel. To demonstrate this property, let/? = the pressure on any float A B perpendicular 



