274 OF THE EQUILIBRIUM OF FLOATATION. 



p i= 2' s" -|- ds, 



and in like manner, the pressure upwards, is 

 p' = Ss' -\-d's. 



But when the body floats in a state of equilibrium, the upward and 

 the downward pressures are equal to one another ; hence we have 



from which, by transposing and collecting the terms, we get 



SY'zzas'-Kd' d)s. 



Now, it is obvious, that what we have demonstrated above with 

 respect to the point D, may also be demonstrated to hold with 

 respect to every other point of the surface whose section is ADC; 

 consequently, by taking the aggregate of the upward and downward 

 pressures, we obtain 



&c.)"=(3 + S + 3+ &c.)s' + {(d' + ef+ d'-f&c.) 



Put w zz (o ; -f- S' + 2' -j- &c.) to infinity, equal to the magnitude 



of the entire floating body, 

 m (3 + I 4- 3 4- &C to infinity, equal to the part immersed 



in the heavier fluid, and 

 m"= {(d' 4- d' 4- d' 4- &c.) (d -f d 4- d + &c.)} to infinity, 



equal to the part immersed in the lighter fluid ; conse- 



quently, by substitution, we get 



ms" = m's' + m"s. (213). 



This equation involves the principle announced in the Proposition, 

 and its application to practical cases will be exemplified in the reso- 

 lution of the following Problems. 



PROBLEM XLIX. 



340. Suppose that a solid body in the form of a regular cube, 

 is observed to float in equilibrio between two unmixable fluids 

 of different specific gravities : 



It is required to determine, how much of each fluid is dis- 

 placed by the body, the specific gravities of the body and the 

 fluid being given. 



Let A BCD be a cubical body, floating in equilibrio between the 

 two unmixable fluids, whose upper horizontal surfaces are respec- 



