1 59-] STABILITY. 95 



Hence, three forces F 15 F 2 , F 3 in the same plane, applied at 

 points Aj, A 2 , A 3 , are in astatic equilibrium if they meet in a 

 point O situated on the circle passing through A lf A 2 , A 3 . 



The condition of equilibrium of the three forces also requires 

 that 



F, F, F. 



sin (F 2 F 3 ) sin (/ r 3 / 7 1 ) sin 



by the property of the circle (Fig. 44), we have ^ (F 2 F 3 ) = A v 

 %.(F 8 Fi) = A 2 , '^.(F 1 F 2 ) = 7r A 3 ; and as the sines of these angles 

 of the triangle A^A^A^ are proportional to the opposite sides, 



we have 



F, F 



1 = A. 



^2^3 ^3^ 



i.e. three forces in astatic equilibrium are to each other as the sides 

 of the triangle formed by their points of application. 



158. The results of the preceding article can be interpreted 

 from a somewhat different point of view. Let two of the forces, 

 F 1 and F 2 , be given, and let it be required to determine their 

 resultant for astatic equilibrium. This resultant F 9 must evi- 

 dently pass through a definite point A 3 of the circle described 

 through the points of application A v A 2 of the given forces and 

 their intersection O. This point A s , through which the resultant 

 must pass, howsoever the two given forces be turned about 

 A v A 2 , is called the centre of the forces. 



If the two given forces be parallel, the point O lies at infinity, 

 and the circle through A I} A 2 , O becomes the straight line 

 A^A^. The point A 3 is therefore situated on this line and 

 divides the distance A-^A 2 in the inverse ratio of the forces F v 

 F 2 , by Art. 157. Compare Art. no. 



159. These results are readily generalised. Any plane sys- 

 tem of forces has a centre unless the resultant be zero. To 

 find the centre we have only to combine the forces in succession, 

 i.e. to find the centre of two of the forces, then the centre 

 of their resultant and a third force, etc. 



