340 OF THE POSITIONS OF EQUILIBRIUM. 



dicularly in f\ then are the triangles tzf and zgv similar to one 

 another, and tf is half the difference of the sides D H and CE ; that is, 



tf-\(x-y}. 



Consequently, by the property of the right angled triangle, that the 

 square of the hypothenuse is equal to the sum of the squares of the 

 base and perpendicular, we shall have 



and by extracting the square root, it is 



But by the property of the centre of gravity alluded to in the con- 

 struction of the diagram, it follows, that 



* + (x-y? : gz : : 3(* + y) : 2 

 or by equating the products of the extremes and means, we get 



and from this, by division, we shall obtain 

 _(<2 



then, because of the similarity of the triangles tzf and zgv, it is 



tz : zf: : gz : gv, 

 or by substituting the analytical expressions, it becomes 



and finally, by working out the analogy, we obtain 



d -W* + y\ 



~ 3(x + y) 



Referring now to the original diagram, or that on which the prin- 

 cipal part of the investigation depends, it will readily appear, that 

 since sr passes through g, the centre of gravity of the quadrilateral 

 figure HECD, it follows, that sg is equal to gr; but by the construc- 

 tion cv is equal to sg, and consequently equal to |sr; now sr is 

 manifestly equal to sw and wr taken conjointly ; therefore we have 

 cvnr \(sw -\- wr). 



Since by the construction, the lines DH and sr are parallel to one 

 another, the triangles HE* and EW are similar; therefore, by the 

 property of similar triangles, we have 



E : tii : : EM; : wr. 



