OF FLOATATION AND THE SPECIFIC GRAVITY OF BODIES. 213 



points G and F ; make ob and F/each equal to /IG or ZF, the perpen- 

 dicular depth of the points G and F below the surface DE; then, 

 according to the principles which we have propounded and demon- 

 strated in the first proposition and its subordinate inferences, the 

 perpendicular pressures upon the indefinitely small portions of the 

 body GH and FI, may be expressed as follows, viz. 



/> zr s XG H X G &, and p' m s X F i X F/, 



where s denotes the specific gravity of the fluid, and p, j/ the respec- 

 tive pressures exerted by it perpendicularly to GH and FI, any 

 indefinitely small portions of the floating body. 



But it is manifest from the resolution of forces, that the pressures 

 of the fluid in the directions bo and y*F, may each be decomposed 

 into two other pressures, the one vertical and the other horizontal ; 

 for by completing the rectangular parallelograms oabc and Tdfe, it 

 is obvious that the pressures in the directions G, CG and dr, ev are, 

 when taken two and two, respectively equivalent to the pressures in 

 the directions bo andy*F. 



Now, the horizontal pressures CD and CF, by construction are equal 

 to one another, and they operate in contrary directions; consequently 

 they destroy'each other's effects, and the upward vertical pressures on 

 the solid at the points G and F, are respectively indicated by the 

 straight lines G and C?F drawn into the specific gravity of the fluid; 

 therefore, the whole vertical pressures on the indefinitely small por- 

 tions GH and FI, are as follows, viz. 



j9 = 5XGHXG, and jy'zzrsXFiX^F, 



where p and p', instead of indicating the perpendicular pressures as 

 formerly, are now considered in reference to the vertical pressures. 



Since the parallel straight lines GF and HI are indefinitely near to 

 one another, the lines G n and FI may be assumed as nearly straight, 

 and consequently, the elementary triangles Giir and FIS are respec- 

 tively similar to the triangles GBO and F/W; therefore, by the pro- 

 perty of similar triangles, we have 



G 6 : G a : : G H : G r, and $f: F d : : F i : F s ; 



and from these analogies, by equating the products of the extreme 

 and mean^terms, we obtain 



G&XGnziGaXGH, an( j F y XFSZHFC? XFI. 



Let therefore, the products G&Xor and F/XFS be substituted 

 instead of GiiXctG and FiXc?r in the above values of p and //, and 

 we shall have 



/; s X G b X G r, and p m s X F/ X F s. 





