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



sumed will bo measured by the mass multiplied into the velocity, and is 

 therefore expressed by 



i 



- V2?H + ni>He 



g (i + K') 



and this subtracted from the whole pressure of reaction, there remains as 

 the whole effective pressure, 



w 



To 



2J*gj- \c(2gH + nv>)-v^2glI + nA 



TVV1 + K' 



And putting for w S, its equivalent T o we have as the pres- 



sure with which the machine moves, 



f===f<|« V2gH + ne»— el 



9V 



This will therefore be a measure of the burthen, p, which the machine 

 can carry at a velocity of v feet per second ; and the pressure multiplied 

 into the space moved through, that is p v, being the measure of the la- 

 bouring force, we have as the expression of the total efficiency of the 

 machine, 



pv 



= {77T=p?'( cV2 * H + "'' 2 - , ')} 



And taking, WH = 1, the whole mechanical value of the water ex- 

 pended, the ratio of the efficiency of the machine will be represented by 

 1 



From this reasoning it appears that the whole pressure expended in giving 

 motion is, 



wS 



i J V2#H + nv 2 \v 



and the entire reaction due to the volume of water expended, being 



1 + K' V T g> 

 which the machine can attain v 

 i, that 



wS c (2H + nJ?L) = -^L= j V2^H + »^) 

 1 + K' ^ g> g^l + K' \ y j 



id resolving this equation, we find as the limit, 



„ = c A / _JzI = cV "^ 5 



V l + K— c*» / _ . ^s 



V l + K-^l-sr) 



the limit of velocity which the machine can attain when moving without 

 burthen, will be such, that 



