OF THE POSITIONS OF EQUILIBRIUM. 315 



) a 2 



_6 2 cV-5) 2 



"I 72 "" (239). 



395. The above is the general equation, whose roots give the several 

 positions in which the solid may float in a state of equilibrium ; it is 

 similar to equation (229), having (5' s) instead of s, and (s' sf 

 instead of s 2 ; the body may therefore have three positions of equili- 

 brium, but it cannot have more, the very same as in the case, where 

 it floated with only one of its edges below the surface of the fluid. 



The method of applying the general equation to the determination 

 Of the positions of equilibrium, is to calculate the value of d, cos.0 

 and cos.^>' from the given dimensions of the section, and to substitute 

 the several given and computed numbers instead of their symbolical 

 equivalents ; this will give a numeral equation of the fourth degree, 

 which may be reduced either by approximation or otherwise, accord- 

 ing to the fancy of the operator. 



396. EXAMPLE. Suppose a solid homogeneous triangular prism, 

 the sides of whose transverse section are respectively equal to 28, 23 

 and 18 inches, to float in equilibrio on a cistern of water with two of 

 its edges immersed ; it is required to determine the positions of equi- 

 librium, on the supposition that the two longest sides of the section 

 include the extant angle, the specific gravity of the prism being to 

 that of water, as 565 to 1000 ? 



In order to resolve this question, we must first of all determine the 

 length of the line cr, which is drawn from the extant angle at c to the 

 middle of the opposite side AB ; for which purpose, let the dimensions 

 of the section be respectively substituted according to the combination 

 exhibited in equation (237), and we shall have 



d = \ V2(28 2 + 23 2 ) 1 8 2 \ v/ 2302 = 23.99 inches nearly. 



Consequently, 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, the elements of Plane Trigonometry 

 supply us with the following equations, viz. 



In the triangle ACF, it is 



and in the triangle BCF, it is 



