OF SPECIFIC GRAVITY AND THE WEIGHING OF SOLID BODIES, 233 



PROBLEM XXXV. 



270. If the weights which a body indicates, when weighed 

 in air and in water, are exactly ascertained : 



It is required from thence to determine the specific gravity 

 of the body, the specific gravities of air and water being 

 known. 



Here also, as in the case of the preceding Problem XXXIV., and 

 Proposition V., the aid of a diagram is not required ; for it would be 

 totally inconsistent with scientific precision, to denote the specific 

 gravities of bodies by geometrical magnitudes, 



Put W i= the real weight of the solid body, 

 w =z the weight when weighed in water, 

 w' zz the weight when weighed in atmospheric air, 

 s =. the specific gravity of water, expressed by unity, 

 s' =. the specific gravity of air, and 

 S = the required specific gravity of the solid body. 



Then, according to the principle announced and demonstrated in 

 the 5th proposition, we have 



W w : W ::s : S ; 



where it is manifest, that W w; expresses the weight which the body 

 loses by being weighed in water ; therefore, we have 

 W (W -w) : W : : Ss : S; 

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



Sw = (Ss)Vf; 



and by proceeding in a similar manner when the body is weighed in 

 air, we obtain 



Now, from the first of these equations, we have 



and from the second, it is 



W = (S=7) ; 



hence by comparison, we obtain 

 Sw Sw' 



