230 OF FLOATATION AND THE SPECIFIC GRAVITY OF BODIES. 



Put m nr the magnitude of the immersed solid, 

 w ~ its weight, 



w"~ the weight lost by the pressure of the fluid, 

 w' the weight of a quantity of the fluid, of the same bulk as 



the solid tody, - 



s the specif of* the solid, and 

 s' in the specific gravity of the fluid. 



Then, because, as we have already stated in a former part of this 

 work, the wejghts are directly as their magnitudes drawn into their 

 respective* gravities ; it follows, that 



A j / / 



w=.ms, and w ~ms ; 



consequently, by analogy, we obtain 

 w' : w : : ms' : ms. 



Now, when a body is immersed in a fluid of less specific gravity 

 than itself, it obviously must descend, in consequence of its superior 

 gravity, and the force of descent is equal to the difference between 

 the weight of the solid body and the weight which it loses by the 

 action of the fluid ; but it has been shown in the preceding propo- 

 sition, that the force of descent is equal to the difference between the 

 weight of the solid, and that of an equal bulk of the fluid ; conse- 

 quently, if f denote the force of descent, then in the one case, we have 



fw w' 9 

 and in the other case, it is 



fw w"; 

 therefore, by comparison, we obtain 



w w' ~ w 10" ; 

 hence, by expunging w, it becomes 



w' w". 



If, therefore, we substitute w" instead of w' , in the first term of the 

 above analogy, we shall get 



w" : w : : ms* : ms : : s' : s ; 

 hence the truth of the proposition is manifest. 



The part of the weight which the body loses by descending in the 

 fluid, is not annihilated, it is only sustained by the upward pressure 

 of the fluid opposing the descent of the body ; and this is the reason 

 why the weight of a vessel full of water is not perceptible while it is 

 beneath the surface. 



