328 OF THE POSITIONS OF EQUILIBRIUM. 



but according to the conditions of equilibrium, these are equal, hence 

 we have 



CD 2 2C D.C F COS.30 = C E 2 2CE.CF COS.30 ; 



therefore, by substituting the analytical expressions, and transposing, 

 we get 



x 3 z/ 2 = 2d cos.30(a? y\ 



and dividing both sides by (x y), we shall have 

 2dcos.30 = a:-f y. 



By Plane Trigonometry cos. 30 nr sin. 60, and by the property of 

 the equilateral triangle, we have e? = b sin. 60 ; consequently, by sub- 

 stitution, we get 



2&sin 2 .60~ x + y; 



or numerically, we obtain 



2X28x1 = 25.954-16.05 42. 



413. Hence it appears, that in so far as the equilibrium of floatation 

 depends upon the equality of the lines FD and FE, the condition is 

 completely satisfied, and the same may be said respecting the lines 



fd and/e ; but it is manifest, that another condition must be fulfilled 

 before the body attains a state of perfect quiescence, and that is, that 

 the area of the immersed part ABED, is to the area of the whole section 

 ABC, as the specific gravity of the solid body, is to that of the fluid on 

 which it floats, or as 0.46875 to unity : now, this condition is evidently 

 satisfied, when 



x y = P(l 0.46875), 



therefore, numerically we obtain 



25.95X16.05 28 2 X0.53125i=41.65. 



Here then, both the conditions of equilibrium are satisfied, and from 

 this we infer, that the positions exhibited in the diagram are the true 

 ones, the downward pressure of the body in that state, being perfectly 

 equipoised by the upward pressure of the fluid. 



414. What we have hitherto done respecting the positions of equi- 

 librium, has reference only to a solid homogeneous triangular prism, 

 floating on the surface of a fluid with its axis of motion * horizontal ; 



* When a solid homogeneous body, in a state of equilibrium on the surface of a 

 fluid is disturbed by the application of an external force, it will endeavour to restore 

 itself by turning round a horizontal line passing through its centre of gravity, and 

 th^s line on which the body revolves, is called the axis of motion. 



