Mr. W. M. Buchanan's Theory of the Jituction Water- Wheel. 127 



which added to the pressure due to tlio actual head, gives as the total 

 effective head, „* . v v „* 



H + i( 1 -<lT>')= H+ "J 



when n is put for 1 — (w)' But u^er tn ^ s new condition, the coeffici- 

 ents *, /3, y, 8, «, will pass to the new values *', £', y', 3', «', and 1 + K will 

 be changed to 1 -f K\ We might proceed to determine the values of these 

 coefficients in terms of « ; but, for our present purpose it is sufficient to 

 observe that they can always be assigned, and may therefore be regarded 

 as known. It will not, however, be out of place to observe that, although 

 *, /3, y, continually increase as v, the coefficient of contraction 8, will 

 decrease till that value of t) is attained, at which the power of the 

 machine is a maximum, and will continually increase afterwards. The small 

 quantity «, may be considered constant ; but, in order to justify ite change 

 to «', we shall suppose it to include the resistance which the atmosphere 

 offers to the motion of the machine ; and which, at very high velocities, 

 becomes an appreciable quantity. 



These conditions being established, we shall have under the increased 

 head-pressure, an increase in the expenditure of water by the machine, as 



/ 2gH /2gH + 



K' 

 and, therefore, in the unit of time we shall have 



Q = S y/ -g^ + ^f cubic feet. 



1 / * 15 s 



And supposing the permanent head H, to become - — ( H 4- n t~) 



by merely elevating the reservoir : it is clear, from what has been before 

 stated respecting the measure of the reaction, that 



This is the measure of the whole pressure of reaction at the orifices; but 

 it is to be observed that the part n ■« — being obtained, in consequence 



of the motion of the machine, with a velocity of v ft. per second, a por- 

 tion of the reaction must have been consumed in communicating that 

 velocity to the volume of water discharged in that second of time, 



equivalent to raisiug it to a height n — - — feet. The pressure thus con- 

 Vol. IT.— No. 2. 4 



