ON PADDLE WHEELS. 155 



to its surface, that is, in the direction F S; the tangential pressure, which is of course in the 

 direction E S perpendicular to C S, is equal to p cos. 8, since the angle E S F = 8, and the 

 moment of resistance due to the given float is equal to the product of this pressure by the 

 circumferential velocity of the spindle, or p V cos. 8. Now the angle G S H being also equal 

 to 8, we have also the effective pressure = p cos. 8, and p v cos. 8 = the moment of effective 

 resistance due to the given float ; therefore, since in any position of the float the effective is to 

 the total resistance as v is to V, or as p to R, the same proportion must hold good for the 

 sum of all the resistances on the different floats through the whole revolution of the wheels. 



The ratio of the speed of the vessel to that of the wheel is even less limited than with the 

 common wheel, or, what is still better, the floats may be made of a greater depth without the 

 upper edge backing water ; for it is evident that when the upper edge of a float enters the 

 water, as at B, if the velocity of the vessel be to that of the spindle as B M to S E, or as S' B 

 to S' S, or putting for the two latter their values, as 2 R cos. 8 f : 2 R cos. 8, the upper edge 

 of the float will move in the direction of its surface, and will consequently have no effective 

 velocity until it has passed that position. Assuming this ratio of velocities, we have 



p: R:: 2 R cos. 8 /: 2 R cos. 8, 

 whence, 



p = R - 9~r h > or /= 2 ( R - ?) cos - 8 - 



COS. O 



If the dip of the float in the middle of the stroke be equal to its depth, cos. 8=1, and we 

 may have 



p = R-L, OT f=2(R-p), 



in which case the rolling circle would extend to J the depth of the float ; and if the velocity of 

 the vessel were to the circumferential velocity of the spindles as 11 to 12, we should have 



R 



7 ' ' ~6 ' 



or the depth of the float one-third of the radius of the wheel. If at the immersion of the 

 upper edge of the float cos. 8 = f , which requires a very great immersion, we should have 



Assuming again p - R, we find 





or the depth of the float = one-fourth of the radius of the wheel. 



Thus, in principle, Oldham's wheel is very superior, but in a practical point of view so 

 exceptionable, that unless some much better method of carrying the principle into execution 

 be devised, it never can be made to succeed. It is, however, perfectly useless to attempt the 

 improvement of this wheel, since the one we are about to describe possesses the same 

 theoretical advantages in a still higher degree, and is as simple in construction as can 

 reasonably be expected. 



