RK> OF FORCES IN ONE PLANE. 



quation to the line of action of the single resultant is 



or, x*Y-y$x=*(Yx- 



x, y being the variable co-ordinates. 



60. The result of the last Article may also be obi. 

 thus. Suppose that the given forces have a single resultant 



: at the point (x, y'}, anl equivalent to the components 

 rallel to the co-ordinate axes. It follows that the 

 given forces will, with A", Y' acting at the p.,int (x, y') 9 

 form a system in equilibrium. Hence, by Art. 57, 



2A- X 1 = 0, 2 F- r = 0, G - Y'x' + Ay = 0. 



Of these three equations the first determines A", the second 

 y, and the third assigns a relation between x and //', 

 is in fact the equation to the line in which the single re- 

 sultant acts and at any point of which it may be supposed 

 to act. If 2 JT and 2F both vanish, it is impossible to lind 

 values of x and y' that satisfy the last equation of the t 

 so long as G does not vanish ; this shews that if the forces 

 reduce to a couple, it is impossible to find a single force equi- 

 valent to them. 



61. In Art. ."i;, we have for the moment of the force P l 

 about the origin the expression 



and this we may express by 



ee X l and Yj are the rectangular com; of P 19 we 



see by comparing the two expressions that the moment of 

 a force about any origin is equal to the algebraical sum of 

 the momenta of its rectangular components about the same 



ich theorems con- 

 nected with moments, and the demonstration of some of them 



