ON THE BOTTOM OF ANY VESSEL. 175 



to the horizon. Then are AB, DC and EF, the respective surfaces of 

 the several fluids, as mercury, water, and olive oil, also parallel to 

 the horizon ; for, as we have elsewhere stated : 



The common surface of two fluids which do not mix, is 

 parallel to the horizon. 



Now, it is manifest, (since the sides A H and B G are perpendicular 

 to the base HG), that the pressure upon the base HG, is equal to the 

 pressures or weights of the several fluids contained in the vessel ; 

 therefore 



Put d zz EH, the perpendicular depth of the lowest stratum EG, 

 d' =. DE, the perpendicular depth of the middle stratum DF, 

 d"=: AD, the perpendicular depth of the upper stratum AC, 

 p zz the pressure of the stratum E G upon the line H G, 

 jt/zz the pressure of the stratum DF upon the line EF, 

 p" the pressure of the stratum A c upon the line D c ; and let s, 



s' and s" denote the specific gravities of the respective 



fluids. 



Then, since the pressure upon any surface, is equal to the area of 

 that surface, drawn into the perpendicular depth of its centre of 

 gravity; it follows, that the pressure upon HG, occasioned by the 

 fluid in EG, is 



^ZZTHGX^S, 

 and in like manner, the pressure upon EF, is 



j/ = EFXdV, 

 and lastly, the pressure upon D c, is 



But the total pressure upon H G, is manifestly equal to the sum of 

 these pressures ; therefore, if P denote the entire pressure on the line 

 H G, we have 

 P 



but the lines HG, EF and DC, are equal among themselves, therefore 

 we get 



P = HG (ds + d's' -f- d"8 u ). (136). 



179. In the preceding investigation, we have considered three fluids 

 of different densities to be contained in the vessel; but the same 

 mode of procedure will extend to any number whatever, and what we 

 have done respecting three fluids is sufficient to discover the law of 

 induction for any other number. It is this : 



