428 Mr. Ivory's Notes relating to 



tive force of the machine, according to the formula (2), is 

 therefore equal to P x v. Now 



F xv = 2avx -£ -2aV x >&.% 

 and, by substituting the value of V 2 , 



?xv=2av(hx-£-)-2a\^-. 



Eider's Formula, 



In the formula (1), put 2 V g (h+f) and 2 s/ gf for V 

 and v; then the impulse to turn the machine will be, 



2arx2g(h+f-s/ hf+f>), 



which is Euler's formula. 



N. B. It must be observed that Euler makes gravity the 

 unit of forces; that is, he denotes it by 1, and expresses the 

 velocities by the square roots of the heights : and, in order to 

 make the foregoing expression agree with Euler's assumption, 

 we must make 2g = 1 ; because 2g has been taken for the 

 measure of the force of gravity. 



Bossufs Formula. 



This author considers the machine in a form more compli- 

 cated than we have here contemplated ; but, when we make 

 allowance for the peculiar mechanism he supposes, the re- 

 sult of his investigation will be found to agree with the ex- 

 pression (1). 



V. Effect of the Machine. 



We have found that the moving power of the machine is 

 Vxv; and 



P x v = v x = aV x — x ( 1). 



Now a X V is the prism of water that issues from the orifices 

 in a second, which quantity we shall denote by Q ; then the 

 equation, 



Q = a x V (A) 



will express the relation between the water expended, the ve- 

 locity of expenditure, and the sum of the areas of the orifices. 

 Further, put 



x - v - V7~ 



and 



