154 SCIENTIFIC DESCRIPTION OF THE HYDROSTATIC WEIGHING MACHINE. 



161. If the fluid be water, whose specific gravity is represented by 

 unity, the equation becomes somewhat simpler ; for in that case, we 

 have 



w' = .7854 &' (D -f d 2 ). (125). 



From this equation the magnitude of the additional weight, or the 

 measure by which it is expressed, can very easily be ascertained ; 

 and the practical rule by which it is discovered, is as follows. 



RULE. Multiply the sum of the squares of the diameters, 

 by .7854 times the rise of the fluid in the tube, or the eleva- 

 tion above the first level, and the product will express the 

 magnitude of the additional weight. 



EXAMPLE. The diameter of a cylindrical vessel is 16 inches, and 

 that of the communicating tube one inch ; now, supposing the machine 

 in the first instance, to be in a state of equilibrium, and that by the 

 addition of a certain weight on the moveable cover, the water in the 

 tube rises 6 inches above the original equilibrating level ; how much 

 weight has been added ? 



By proceeding according to the rule, we have 

 D 2 -|- d*= 16 2 + P i=256 + 1 = 257, 



and by multiplication, we obtain 

 w' = .7854X6X257 zz 1211.0868 avoirdupois Ibs.* 



162. If the additional weight, by which the water is made to rise 

 in the tube be given, the distance above the first level to which it will 

 rise, can easily be found ; for let both sides of the equation (125), be 

 divided by the quantity .7854 (D* -\- d z ), and we shall obtain 



,,_ 



~.7854(D*4-d 2 )' 



And from this equation, we deduce the following rule. 



RULE. Divide the additional weight, by the sum of the 

 areas of the moveable cover and the cross section of the 

 communicating tube, and the quotient will give the height to 

 which the fluid will rise above the first level. 



* It is manifest from the form of the equation which supplies the rule, that 

 without paying particular attention to the nature of the load which produces the 

 equilibrium in the first place, the value of vf is ambiguous, and may be read in 

 ounces, Ibs., cwts., or tons ; and indeed, in any denomination of weight whatever ; 

 but it must always be read in the same name as that by which the equilibrium 

 is produced. 



