OF THE POSITIONS OF EQUILIBRIUM. 341 



or by substituting the analytical expressions, it becomes 



from which, by working out the proportion, we get 



tt . r -(* y)(2* + y). 



**9] 

 consequently, by adding and dividing by 2, we obtain 



6(0: + y) 



Again, it is obvious by the construction, that the triangles HE* and 

 vcd are similar to one another ; hence we have 



HE : ut : : cv : vd; 

 but by the property of the right angled triangle, it is 



HE ^/b*+(x yf; 

 consequently, by substitution, we obtain 



hence, by reducing the proportion, we get 



V 



It is furthermore manifest, that the triangles HE* and vgn are 

 similar to one another ; consequently, we have 



HE : E * : : gv : gn\ 

 or by substituting the respective values, we get 



from which, by reduction, we obtain 

 ) 



~ 



but ge = yn 4. en ; therefore, by addition, we have 



(* + y) 



(266). 



428. This is one side of the equation which involves the second 

 condition of equilibrium, and in order to determine the other side, we 

 must have recourse to the triangles acb and a cm, which together 



