OF THE HYDROSTATIC WEIGHING MACHINE. 153 



Then it is manifest, that when the equilibrium originally obtains ; 

 that is, when the surface of the fluid in the tube is at a, and that in 

 the vessel at EF, the pressure of the fluid in the tube exerted at b, is 



where the symbol p denotes the pressure at b ; 



but this is manifestly in equilibrio with the pressure of the column 

 wEFx t or the weight w ; consequently, we have 



.7854rf 2 As : w : : .7854tf : .7854D 2 ; 

 or by suppressing the common factors, we obtain 



hs : w:: 1 : .7854 D 8 ; 

 and by equating the products of the extremes and means, it is 



wn=.7854sD 2 . (121). 



Again, by means of the additional weight w', whose magnitude is 

 required, the cover EF is supposed to descend to the position mn; 

 while, in order to regain the equilibrium, the fluid rises in the tube as 

 far as the point K, in which case, the altitude of the equilibrating 

 column CK becomes (h -f- ti -f- 5) and consequently, the pressure at 

 c, is 



;>'=.7854eZ 2 s (& + &' + 3), 



and this is in equilibrio with the pressure of the column ymnz, or 

 the weight (w + w') ; consequently, we have 



.7854d 2 s (h + h' -f 3) : w -f w' : : .7854d 2 : .7854 D * ; 

 or by suppressing the common factors, we have 



s (k-\-h' -M) : w + w' : : 1 : .7854o 2 ; 



therefore, by equating the products of the extremes and means, we get 

 w -\- w/zz .7854D 2 s(/z- r -A'- r -S). (122). 



But we have seen above, equation (121), that wnr.7854/i5D 2 ; 

 consequently, by substituting and separating the terms, we obtain 



W /i=.7854D 2 s(A / + a). (123). 



Now, it is manifest, that the descent of the cover in the vessel, and 

 the rise of the fluid in the tube, must be to one another, inversely as 

 the squares of the respective diameters ; therefore, we have 



Ztf^h'tf, 



* or by division, we get 

 h'd* 



S = l^' 



and finally, by substitution, we obtain 



w' = .7854 h's (D 2 -f d 3 ). (124). 



