Mr. W. M. Buchanan's Theory of the Reaction Water-Wheel 129 



when n is replaced by its equivalent 1 — /JL\ 



From this, it appears that the maximum head-pressure which can be 



c 2 H 



created in the machino by centrifugal force = — - 



l + K-vd-sr 



and tho ratio 



VfyH 



>\/l + KW(l-J) 



A question of much more importance in practice is the value of v t 

 which will render 



a maximum. But at this point calculation fails to be satisfactory, and 

 we must have recourse to experiment to determine the relation which v 

 bears to the velocity V 2g H of the water duo to the initial head H. 

 Euler in his elaborate investigation of the general problem, (Berlin, Trans. 

 1750 and 1754,) misled by the symbols resulting from his attempt to deter- 

 mine the maximum value of the function analytically, announced the 

 hypothesis that the power of the machine increases with the velocity ad 

 infinitum. In taking experiment as the guide, it did not, however, require 

 any lengthened investigation to discover that the symbols of calculation 

 do not in this case represent the truo conditions of the question, a nd that 

 v has in practice a limit which can be represented in terms of V 2g H, 

 and of the coefficients K/ and n. In order to ascertain that relation with 

 the necessary degree of exactness, a series of values of p v, taken near the 

 maximum, was interpolated by La Grange's theorem, and the value of v 

 thereby determined, was found to have the relation 



•V T 2 ' h - 



4K 2 



+ K' 



wluch in practice does not differ materially from V 2g H. If then we sub- 

 stitute that value in the expression for p v, we find after reduction, 



pt) = Wx 2H 



( / 5"? 1 



8223 WH 



r 3 1 1 



when -pa = -jr and K' = yrr, and c = 1*05 as before determined. The 



highest value obtained experimentally under circumstances which admit- 

 ted of positive accuracy in the measurement of the water, was 

 •80375 W H ; but in that instance the fall was variable, and the proper 

 velocity could not be maintained during the time of an experiment. It 



