OF THE POSITIONS OF EQUILIBRIUM. 295 



of the transverse section are represented by large numbers ; in parti- 

 cular cases, the ultimate form will admit of being modified, and may 

 in consequence, be rendered somewhat more simple ; but it must 

 nevertheless be understood, that whenever the position of equilibrium 

 is required by computation, we must inevitably perform a very irksome 

 and laborious process. 



A geometrical construction may also be effected by the intersection 

 of two hyperbolas ; but since this implies a knowledge of principles 

 higher than elementary, we think proper to pass it over, and proceed 

 to illustrate the application of the above equation by the resolution of 

 a numerical example. 



375. EXAMPLE. Suppose a triangular prism of Mar Forest fir, 

 the sides of whose transverse section are respectively equal to 28, 26, 

 and 18 inches, to float in equilibrio in a cistern or reservoir of water, 

 having only one angle immersed ; it is required to determine the posi- 

 tion of equilibrium, on the supposition that the two longest sides of 

 the section penetrate the fluid, the specific gravity of the prism being 

 to that of water as 686 to 1000 ? 



By recurring to equation (229), and comparing its several consti- 

 tuent quantities with the parts of the diagram to which they respec- 

 tively refer, it will readily appear, that x, cos.0 and cos.0' are the only 

 terms whose values require to be calculated ; of which cos.0 and cos.0' 

 are to be determined from the nature of the figure, and x from the 

 resolution of the biquadratic equation in which its values are involved. 



The length of the straight line CF, which is drawn from the vertex 

 of the section at c, to the middle of the opposite side at F, is accord- 

 ing to equation (226), expressed by 



consequently, by substituting the numerical values of the sides, we 

 obtain _ 



d= 1^2(28* + 26 2 ) 18 2 = 25.4754784 inches. 

 Hence, in the triangles ACF and BCF respectively, we have given 

 the three sides AC, AF, FC and BC, BF, FC to find COS.ACF and cos. 

 BCF; for which purpose, we have the following equations as deduced 

 from the elements of Plane Trigonometry, viz. In the triangle ACF, 

 it is 



4 c d a 2 



and in the triangle BCF, it is 



