96 STATICS. [160. 



It has been shown in Arts. 141 and 144 that a plane system 

 of forces whose resultant does not vanish can always be reduced 

 to a single resultant R whose line is called the central axis of 

 the system. It appears now that if the forces be all turned by 

 the same angle 6 about their points of application, the line of the 

 resultant, or the central axis, will turn about a certain fixed 

 point called the centre of the system. " For a system of parallel 

 forces the existence of such a centre has already been proved in 

 Art. 1 10. 



160. Analytically, the centre of a plane system of forces is 

 found as the intersection of the two positions of the central axis 

 before and after any displacement of the plane figure, or body, 

 on which the forces act. 



By Arts. 144 and 145 the equation of the central axis is 



^Y>Z-^"n-^(xY-yX}=o. (i) 



Let the figure with the axes of co-ordinates be turned through 

 an angle about an axis through the origin perpendicular to its 

 plane, while the forces keep their original directions. The cen- 

 tral axis of the forces in the new position will have an equa- 

 tion of the same form as before, in which, however, x, y, f, 77 are 

 referred to the new system of co-ordinates. To find the equa- 

 tion of the central axis in the old co-ordinates, we have to 

 substitute x cos </> y sin </> for x, ;rsin (ft+y cos forj>, and 

 similarly for f, 77. This gives 



sn 

 2 \(x cos <f> y sin <) Y (x sin <f> +y cos <j))X ] = o, 



or collecting the terms containing cos <f> and sin <, respectively, 



0=0. (2) 



The centre being the intersection of the lines (i) and (2), its 



