240 OF SPECIFIC GRAVITY AND THE WEIGHING OF SOLID BODIES* 



PROBLEM XXXVIII. 



286. If two bodies of different, but known specific gravities, 

 equiponderate in a fluid of given density : 



It is required to determine the ratio of the quantities of 

 matter which they contain. 



Put 5 the specific gravity of the fluid, in which the bodies are 

 found to equiponderate, 



m zz the magnitude of the greater body, 



s' zz its specific gravity, 



m'zz the magnitude of the lesser body, 



s" zz its specific gravity, 



w zz the weight of the greater body in the fluid, and 



w/zz the weight of the lesser body under the same circum- 

 stances. 



Then, by Proposition V., when a solid body is immersed in a fluid 

 of different specific gravity, the weight which it loses, is to its whole 

 weight, as the specific gravity of the fluid, is to the specific gravity of 

 the solid ; it therefore follows, that 



/ : s : : m s' : m s zz the weight lost by the greater body ; 

 but the weight of the body in the fluid, is manifestly equal to the 

 difference between its absolute weight, and that which it loses in 

 consequence of the immersion ; hence we have 



w zz m s' m s zz m (s' s) ; 

 and by a similar mode of procedure, we obtain 

 s" : s : : m 1 s" : m' s zz the weight lost by the lesser body; 

 consequently, the weight which it possesses in the fluid, is 



w' zz m 1 s" m' s zz m' (s" s). 



Now, according to the conditions of the problem, these are in 

 equilibrio with one another; therefore by comparison, we have 



m (s' s) zz m'(s" s), 

 and by converting this equation into an analogy, it is 



m : m' : : (s" s) : (s' s); 



and finally, if we multiply the first and third terms by s', and the 

 second and fourth by s", we shall have 



ms' : m's 11 : : s'(s" s} : s"(s's). 



287. EXAMPLE. Twenty ounces of brass, whose specific gravity is 

 eight times greater than that of water, and a piece of copper whose 



