Ma)tclicstci' Memoirs, Vol. xliv. (1899), No. 5. 21 



of this action is easily given and is of interest, not only 

 in the application to the horizontal propulsion of an 

 aeroplane, but also because a screw rotating about a 

 vertical axis may be used for direct maintenance. The 

 latter question may conveniently be considered first. 



The screw is su^jposed to maintain a weight W at a 

 fixed position in still air. This it does by creating a down- 

 ward current of velocity v . If 5' be the area of section 

 of the current, equal to that swept through by the screw, 

 the volume of air acted upon per second is S'v, and 

 the momentum generated per second is S'v . pv, or S\)v". 

 Hence 



lV=S'pv" (29). 



Again, the kinetic energy generated per second is ^S'pv'; 

 so that if [/ be the velocity at which [P would have to be 

 lifted to do a corresponding amount of work, we may, 

 neglecting frictional losses, equate the above to UIV. 

 Thus 



l7W=hS'pv' (30). 



From (29), (30), U'= U, 



iuid .S =--jj, (30- 



So far as these equations are concerned, any weight can 

 be maintained by a limited expenditure of work, but the 

 smaller the power available the larger must be the section 

 of the stream of air, and consequently of the screw, or 

 other machinery, by which the air is set in motion. 

 Again from (31) 



IF17= — ^^ (32), 



{4,,S-)l 



so that if S' be given, the whole power required varies 



as W"-. 



To obtain numbers applicable to the case of a man 



