OP THE POSITIONS OF EQUILIBRIUM. 297 



equal to 22 and 22.7 inches, and through the points D and E, draw 



the straight line IK, which will coincide 



with the plane of floatation, or the sur- A 



face of the fluid on which the body floats. \ Jx^xT; -3 



Bisect A B, the extant side of the sec- T \^ ^^1 



tion in the point F, and join F D and F E ; 

 then, the conditions of equilibrium ma- 

 nifestly are, that the lines FD and FE 

 are equal to one another, and that the 

 area of the immersed triangle DCE, is 

 to the area of the whole triangle ACB, 

 as the specific gravity of the solid is to 

 the specific gravity of the fluid. 



That the lines FD and FE are equal to one another, appears from 

 an inspection and measurement of the figure ; but the following proof 

 by calculation will be more satisfactory, inasmuch as numbers can be 

 more correctly estimated than measured lines, which depend for their 

 accuracy upon the delicacy of the instruments and the address of the 

 operator. 



In the plane triangle DCF, we have given the two sides DC and CF, 

 respectively equal to 22 and 25.4754784 inches, and the natural cosine 

 of the contained angle DCF equal to 0.94769 ; consequently, the third 

 side D F can easily be found ; for by the principles of Plane Trigono- 

 metry, we know that 



DF 2 =:DC 2 4- FC 2 2DC.FCCOS.DCF; 



therefore, by substituting the respective numerical values, we obtain 



DF 2 = 484 + 649 2X22X25.4754784X0.94769 = 70.72; 



consequently, by extracting the square root, it is 



DFZZI V 7 0.72 = 8.4 inches. 



Again, in the plane triangle ECF, we have given the two sides 

 EC and CF, respectively equal to 22.7 and 25.4754784 inches, and 

 the natural cosine of the contained angle ECF equal to 0.93906; 

 consequently, by Plane Trigonometry, we have 



EF 2 zrEC 9 -|-CF 2 2EC.CFCOS.ECFJ 



and substituting the respective numerical values, we obtain 

 EF 2 =: 515.29 -f- 649 2 X22.7 X25.4754784 X 0.93906 =: 77.895 ; 

 therefore, by extracting the square root, we shall have 

 E F zz: V 777895 8 .82 inches. 



