302 OF THE POSITIONS OF EQUILIBRIUM. 



Expressions of this form, arising from the reduction of an adfected 

 quadratic equation, are in general rather troublesome and difficult to 

 render intelligible in words, and even when intelligibly expressed, they 

 are to say the least of them, but very dull and uninviting guides, from 

 which a tasteful reader turns with disgust ; we are therefore unwilling 

 to crowd our pages with long and formal directions for the purpose of 

 reducing equations, when it is probable after all, that nine out of every 

 ten of our readers will pass them over, and proceed immediately to 

 discover their object by the direct resolution of the original equation. 



384. It is however necessary, in conformity to the plan of our 

 work, to express the most important final equations in words at length, 

 and since the preceding forms are of considerable utility in the doctrine 

 of floatation, it would be a direct violation of systematic arrangement, 

 to omit the verbal description, and leave the subject open only to 

 algebraists ; we shall therefore, in order to render both parts of the 

 operation intelligible, endeavour to express the method of reduction 

 in as brief and comprehensive a manner as the nature of the subject 

 will admit. 



1 . To determine the value of x. 



RULE. From four times the square of one of the equal 

 sides of the section, subtract the square of the base, or side 

 opposite to the vertical angle ; multiply the square root of the 

 remainder by one half the natural cosine of half the vertical 

 angle, and call the product ?n. 



From four times the square of one of the equal sides of the 

 section, subtract the square of the base, or side opposite to the 

 vertical angle, and multiply the remainder by one fourth of 

 the square of the natural cosine of half the vertical angle, 

 or that which is immersed in the fluid ; then, from the product, 

 subtract the quotient that arises, when the specific gravity of 

 the solid, drawn into the square of one of the equal sides 

 of the section, is divided by the specific gravity of the fluid, 

 and call the square root of the remainder n. 



Finally, to and from the quantity denoted by m, add and 

 subtract the quantity denoted by n ; then, the sum in the one 

 case, and the difference in the other, will give the two values 

 of*. 



2. To determine the corresponding values of y. 



RULE. Calculate the values of m and n, precisely after the 

 manner described above; then, from and to the quantity 



