Principle of the Screw to the Floats of Paddle-wheels. 259 



The moment 



= ip i co?xr 4 X'13l4 i . 



= iW^ 5 .Axi{l-(A) 4 } 

 -\pf»?r* x -113985. 



On comparing these results, we see that when the two wheels 

 are revolving with the same angular velocity, the resistance and 

 moment on the new float are less than on the old, but the ratio 

 between the work done and the power required to perform it is 

 greater in the case of the new wheel than in that of the old, in 

 the ratio of 7 to 6; so that the advantage is on the side of Dr. 

 Croft's invention. If the difference of the powers of the two be 

 spent in increasing the angular velocity of the new wheel, then 

 the resistance will be greater in the new wheel than in the old, 

 in the ratio of 7 to 6 ; and the angular velocities in this case 

 would be in the ratio of 6 to 5. 



The mathematical theory takes no account of the motion of 

 the water ; so that the results of the above calculations are quite 

 separate from the other evident advantages stated in the early 

 part of this paper. Also, since the floats enter the water gradu- 

 ally, and since the water is less disturbed, it is plain that the 

 angular velocity of the new wheel may be increased in the above 

 ratio without causing the floats to slip through the water and 

 do no work. 



Another case, where the results may be more easily obtained 

 from the expressions after the first integration, is that in which 

 the axis of the wheel is the axis of the screw-surface, and where 

 the inner portion of the surface up to a distance of one-third of 

 the radius from the axis is cut out, so that a=0, and the limits 



v 



of integration for a are r and 5 . Hence 



R=ip / a,V|/;(2-f)cos^^(9~2/ Y^--tan- 1 i)cos^6'} 



=i/o,©Vxsin0x-4O6, 

 and 



very nearly. 



