ON PADDLE WHEELS. 



147 



proportions of the parts cause it to change its position ; indeed it would be impossible to 

 make the wheel revolve, if any other length or position were given to the crank or lever S G. 

 This essential property of Buchanan's wheel is amply sufficient to distinguish it from another, 

 known by the name of Morgan's wheel, which we shall treat of in a future part of this paper. 

 We mention this because some persons have confounded these two wheels together from 

 ignorance of their respective properties, although they are widely different both in principle 

 and construction. 



Fig. 2. PL LXXVI. shows the path of one of the floats of the wheel represented in Fig. 1, 

 under the same circumstances as those of the ' Phoenix' in Fig. 2. PL LXXIV. Fig. 3 is the 

 enlarged node. Figs. 4 and 5 correspond with Figs. 4 and 5, PL LXXIV., which relate to the 

 trial of the ' Salamander.' In both these cases the floats back water, or move forward in the 

 same direction as the vessel, but most considerably at the deep immersion. We will not, 

 however, investigate the action of the wheel under such circumstances, but suppose the 

 velocity of the wheel to be such that the floats, at their immersion and emersion, shall move in 

 a vertical direction. 



In the adjoining figure let C S represent the radius, A B the float, and S the spindle, at 

 the instant the lower edge of the float enters the water, C D a vertical line, A D the water 

 line, and S H a horizontal line, passing through 

 the spindle S and intersecting the line C D 

 at the point H. Let C S = R, C H = a, A B =/, 

 and the breadth of the float = b ; let $ = the in- 

 clination of the radius at any given instant, and 

 its inclination at the instant its lower edge enters 

 the water, or the angle S C D, = a ; also let S E, 

 perpendicular to the radius C S, = the circum- 

 ferential velocity of the float, = V, and v = the 

 velocity of the vessel. 



In order that the above-mentioned condition 

 may be satisfied, it is evident that there must 

 exist the following relation between v and V, 



V V COS. a. ; 



for v must be equal to A E, which is equal to S E 

 cos. SEA; but the angle S E A = the angle 

 S C D = a. ; therefore v = V cos. a ; and p, the 

 radius of the rolling circle, = R cos. a = a. 



The effective velocity of the float in any given position is V cos. $ v. For let A' B' be 

 the given position of the float, the angle S' C D being equal to 0, and let S' E', perpendicular 

 to the radius C S', = V, and the portion E' G of the horizontal line E' F, which intersects 

 A' B' at the point F, = v ; the straight line S' G will represent the motion of the float both 

 in direction and velocity, and F G, which is perpendicular to its surface, will be equal to its 

 effective velocity. Now we have E' F = S' E' cos. S' E' F = V cos. 0, and E' G = v, there- 

 fore effect, vel. = F G = E' F E' G = V cos. <f> v, or, substituting for V and v their 



respective values, viz., V= 



r Rn 

 30 



and v = Fcos. a = 



rRn 

 30 



cos. a = 



TT a n 



~sd~' 



