OF SPECIFIC GRAVITY AND THE WEIGHING OF SOLID BODIES. 253 



Put m iz: the magnitude of the solid, 



s ~ its specific gravity, or the quantity which the problem 



demands, and 

 s' zr the specific gravity of air at the time of the experiment. 



Then, because the weight of any body, is expressed by the product 

 of its magnitude drawn into its specific gravity ; it follows, that when 

 the weight of the air is disregarded, the weight of the solid is 



rv' zn m s ; 



but the weight of a quantity of air equal in magnitude to the body, 

 is ms', and it evidently loses as much weight as that of the fluid which 

 it displaces ; consequently, when the weight of the air is considered, 

 the weight of the body when weighed in air, becomes 



rv' m m (s s'), 



therefore, by division, the magnitude of the solid is 

 w' 



consequently, the weight in vacuo, is 

 snf 



s s ' 



Then, because rv" denotes the weight of the vessel with the solid in 

 it, when filled up with water, and rv the weight of the vessel when full 

 of water ; then w" rv expresses the weight which the vessel has gained 

 by the immersion of the solid, and this is manifestly equal to the dif- 

 ference between the weight of the solid, and that of an equal bulk of 

 the fluid ; therefore, the weight of a quantity of water, equal in bulk 

 to the solid, is 



sw s(rv -\-w' n")-\-s'(w" rv) 



-. rv -4- rv 7 



s / s / 



Then, as the specific gravity of the body, is to the specific gravity 

 of water, so is the weight of the body, to the weight lost ; that is, 



sw s(w-\- rv' rv") + s' (w" w) 

 : 7=^ : s s' 



and this, by suppressing the denominator in the homologous terms, 

 becomes 



s : 1 : : sw : s(rv-\-rv' rv")-\-s'(w" rv), 



and by equating the products of the extreme and mean terms, we 

 obtain 



s(w 4- rv' rv") -\- s' (rv" rv) rv' ; 



