342 OF THE POSITIONS OF EQUILIBRIUM. 



with the triangle HE*, are similar among themselves; consequently, 

 we have first, from the triangles list and ac, as follows. 



H E : ii t : : a c : a b ; 

 which by substitution becomes 



y):(* y):: Ja:a6 

 therefore, by reduction, we obtain 



a(x y) 

 - v yj - 



Again, from the triangles HE and aowt, we get 



HE : E : : ao : om; 

 consequently, by substitution, it is 



from which, by reduction, we obtain 

 but GC Gw-f-^c; therefore, by addition, we have 



GC. 



429. Here then, we have discovered the other side of the equation 

 which involves the second condition of equilibrium, and consequently, 

 we are now prepared to determine the positions which the body assumes 

 when floating in a state of rest ; for which purpose, let the equations 

 (266 and 267) be compared with each other, and we shall have 



Here it is manifest that the equation involves two unknown quan- 

 tities ; in order therefore to render it capable of solution, one of those 

 quantities must be eliminated, and this can very easily be done by 

 means of the equation (265), where we have 



therefore, by transposition and division, we get 

 2as 



y 7 x > 



or by supposing s' equal to unity, as is the case with water, we have 

 t/:n2as x. 



