OF THE POSITIONS OF EQUILIBRIUM. 305 



That the positions here exhibited are those of equilibrium, is very 

 easy to demonstrate, for produce the sides CA and cb to meet the 

 surface of the fluid in the points E and D, and bisect AB and a b in 

 the points F and/; then, if the straight lines FE, FH and/D, /i be 

 drawn, they will be equal among themselves. 



This is one of the conditions of equilibrium, as we have already 

 demonstrated in the construction of the original diagram, and the 

 other condition is, that the areas of the immersed figures ECH and 

 DCI, are respectively to the whole areas ABC and a be, as the specific 

 gravity of the solid, is to the specific gravity of the fluid which sup- 

 ports it. 



Now, if the first of these conditions obtain, that is, if the straight 

 line FE be equal to FH, and/D equal to/i, then, by the principles of 

 Plane Trigonometry, we shall have 



EC 2 -j-CF 2 2EC.CFCOS.ECFZTHC*-}- c ^ 2HC.CF COS.FCH ; 



but the angles ECF and FCH are equal to one another, and each of 

 them equal to ; consequently, by substituting the literal representa- 

 tives, we have 



x a -f- d* 2rf x cos.^> y 3 - -)- d* 2dy cos.0, 

 or by expunging the common term e? 4 , we get 



x* 2dx cos.0 z= ?/* ^dy cos.0, 

 and this, by transposing and collecting the terms, becomes 



# 2 ?/ 2 zz: 2e? cos.^(o? y) ; 



therefore, jf both sides of this equation be divided by the factor (x y), 

 we shall obtain 



x -f- y zr: 2rf cos.^>. 



Now, by a previous calculation we found x to be equal to 32.106 

 inches, y 16.752 inches, d equal to \/28 8 10% and cos. equal to 

 0.93406 ; consequently, by substitution, we have 



32.106 + 16.752 = 2x0.93406X6/19"; 

 hence the equality of the lines FE and FH is manifest. 

 What we have shown above with respect to the triangle ABC, may 

 also be shown to obtain in the triangle a be, the one being equal and 

 subcontrary to the other ; this being the case, it is needless to repeat 

 the process; but we have yet to prove, that the area CEH, is to the 

 whole area ABC, as the specific gravity of the floating body, is to that 

 of the fluid on which it floats. 



therefore, by the principles of Plane Trigonometry, we get 



AC : AF : : rad. : sin.ACF, 

 VOL. i. x 



