OF THE EQUILIBRIUM OF FLOATATION. 279 



If any body float upon the surface of a fluid in vacuo, and 

 air be admitted, the body will ascend higher above the surface, 

 and consequently, the proportion of the immersed part to the 

 whole will be diminished. 



PROBLEM LI. 



350. Suppose a solid body to float in equilibrio on the surface 

 of water, both in air and in a vacuum : 



It is required to determine the ratio of the parts immersed 

 in the water in both cases. 



Put m zz: the magnitude of the whole floating body, 



m' zr the magnitude of the part immersed below the surface 



of the water, when the incumbent fluid is air, 

 m" the portion immersed when the body floats on water in 



vacuo, 



s zz: the specific gravity of air, 

 s' zz: the specific gravity of water, and 

 s" zz: the specific gravity of the floating body. 

 Then we have m m', for the part above the surface of the water, 

 when the incumbent fluid is air, and m m" for the extant part when 

 the body floats in vacuo ; consequently, by equation (213), we have, 

 when the body floats in air, 



ms" zz: m' s' -\-(m ra') s, 

 from which, by a little reduction, we obtain 



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

 and finally, by division, it becomes 



,_*(*"-*) 



- (s'-s) ' (217). 



and again, when the body floats in vacuo, we have 



but in this case, s vanishes, hence we get 



ms" = m"s, 

 and by division, it is 

 _ms" 



(218). 



Let the equations (217) and (218) be compared with one another in 

 the terms of an analogy, and we shall have 



' - m ( s '' s ) . . a . ms " . 



' (S' 5) S' 



