number of lbs. per second flowing through a narrow ring on the 

 screw disc, between radii r and r-\-dr is equal to 2irvrpdr, where 

 p stands for the weight of a cubic foot or other standard of 

 length ; consequently the work done, in foot lbs. per second, 

 between radii r and r + dr is 



7r. r 2 0)J2 2 7TO 



dP=27rvrpdrX = llvr^udr. 



g g 

 and the total horse power, one horse power being 550 ft. lbs. per 

 second, is 



H.P=—^- fvr s o>dr 

 S$°gJ 

 the limits of the integral being outer radius and radius of boss. 

 If we now knew the values of v and a> at each point on the screw 

 disc, we could find the value of this integral. If furthermore 

 ,we could exp^^ss v and a in terms of r algebraically in certain 

 known forms, we could calculate the power by the rules of 

 arithmetic, and if we could not express v and w in terms of r in 

 such a way, still we could draw curves of their values, and find 

 the value of the integral by the aid of a planimeter. 



In the case of a turbine matters are much simpler, for v and w 

 are, if not exactly, very nearly, constant, so that the troublesome 

 part of the process is avoided, for the most part, otherwise the 

 case is the same as that of the propeller, only of course the work 

 is done by the water on the turbine, instead of by the turbine, 

 otherwise called the propeller, on the water. In this last respect 

 the difference may be paralled by the work done in the case of 

 a truck running down an inclined plane, and doing work while 

 so doing, which work is the same as that required to pull the 

 same truck up an inclined plane ; and there is nothing novel, 

 or contrary to any previously known or accepted doctrine, or to 

 the facts of experience, in the application made above of the 

 theory of turbines to the case of propellers ; but on the contrary, 

 all who have said anything on the point at all, have described 

 propellers as, in effect, turbines driving water, instead of driven 

 by it, and only differing from ordinary turbines in not having 

 guide blades to give the entering (or in propellers the outflowing) 



