Mr. W. M. Buchanan's Theory of the Reaction Water- Wheel. 123 



tho channel at a distance x from its origin. But the unequal section of 

 the channels renders it impossible to assign exactly tho value of the 



integral / — L d%. This is of little moment, as the quantity itself being 



J K, 



very small, will not diner sensibly from the mean of the resistance which 

 tho water would encounter in passing through the machine if the areas 

 had throughout their length tho mean cross-sectional area of the orifice 

 and origin, and may therefore be expressed by 



u, being the corresponding velocity of the water in channels of the mean 

 transverse area assumed. 



These are direct and evident causes of loss of head-pressure in the 

 machine ; but we have yet to take into account another influence small 

 indeed in amount, but still appreciable in its effect. This is mani- 

 fested in what is denominated the contraction of the vein. Revert- 

 ing to our suppositious example of an orifice being formed in the side of 

 a vessel in which the water is maintained at a constant level, it was left 

 to be inferred that the velocity of efflux would be that of a heavy body 

 falling through a space equal to the head of water ; and if the orifice be 

 very small, compared with the horizontal area of the vessel, this will be 

 nearly true. And, in general, the velocity of discharge can be closely 

 assigned when the ratio of these quantities is known. But although the 

 velocity with which the particles of the fluid issue may be found from data, 

 which are always attainable, it does not follow that we thereby know the 

 volume of liquid discharged. Although invariably proportional to the 

 area of the orifice and to the square root of the head of water, its 

 value is not found to depend, except in a minor degree, upon the ratio 



, but upon the form of orifice through which the jet issues. 



area of vessel 



If the side of the vessel in which the orifice is made, be of very thin 



material, as tin-plate, the discharge q, in cubic ft. per second, will be very 



nearly expressed by 



q = -625aV r "2?H 



in which a is the area of the orifice, and H the head of water under 



which the discharge takes place. 



If the jet from an orifice of this kind be closely observed, it will be 

 perceived to converge through a short distance from its origin, forming, 

 when the orifice is circular, a conoid, of which the area of the least 

 section is £ of the area of the orifice. If advantage be taken of this 

 circumstance to apply an ajutage to tho orifice of the form assumed by 

 the jet, tho discharge will be found to approximate very closely to that 

 assigned by the formula q = a V 2g H. 



This difference of discharge of the two kinds of aperture, is usually 



