OF THE POSITIONS OF EQUILIBRIUM. 293 



zontal surface of the fluid, or the plane of floatation passing through 

 DE; consequently, F H is also perpendicular to DE, and FD, FE are 

 equal to one another. 



Put a A B, the unimmersed side of the triangular section, 

 b nz BC, one of the sides which penetrate the fluid, 

 c zz AC, the other penetrating side, 

 d =2 CF, the distance between the vertex of the section, and 



the middle of the extant side, 

 zz the angle ACF, contained between the side AC and the 



line CF, 

 ^' zz the angle BCF, contained between the line CF and the 



side B, 



s zz the specific gravity of the solid body, 

 s' zz the specific gravity of the fluid on which it floats, 

 x zz CD, the immersed portion of the side AC, and 

 y zz CE, the immersed portion of the side BC. 



Then, according to the principles of geometry, since the line CF is 

 drawn from the vertex of the triangle at c, to the middle of the base 

 or opposite side at F, it follows, that 



AC 2 -f BC 2 ZZ2(AF 2 -f CF 2 ), 



or by taking the symbolical representatives, we shall obtain 



from which, by reduction, we get 



dzziV2(^4-c 2 ) a*. (226). 



Since all straight lines drawn parallel to the axis of the prism are 

 equal among themselves ; it follows, that the weight of the whole solid 

 ABC, and that of the portion DEC below the plane of floatation, which 

 corresponds to the magnitude of the fluid displaced, are very appro- 

 priately represented by the areas drawn into the respective specific 

 gravities of the solid and the fluid on which it floats. 



Now, the writers on the principles of mensuration have demon- 

 strated, that the area of any right lined triangle : 



Is equal to the product of any two of its sides, drawn into 

 half the natural sine of their contained angle. 



Therefore, if we put a' and a" to represent the areas of the triangles 

 ABC and DEC respectively, we shall have for the area of the triangle 

 ABC, 



