64 STATICS. [in. 



The force F 1 + F 2 combines with F 3 to form a resultant 

 .FI + FZ + FP whose point of application x" , y n , z" is given by 



and similar expressions for y n , z n . 



Proceeding in this way, we find for the point of application 

 (x, ~y, z) of the resultant of all the given forces 



with corresponding equations for y and z. We may write these 

 equations in the form : 



As these expressions for x, y, z are independent of the direc- 

 tion of the parallel forces, it follows that the same point (x, y, z) 

 would be found if the forces were all turned in any way about 

 their points of application, provided they remain parallel. The 

 point (x, y, z) is for this reason called the centre of the system 

 of parallel forces. It is nothing but what in geometry is called 

 the mean point, or mean centre, of the points of application if 

 the forces are regarded as coefficients or "weights" (in the 

 meaning of the theory of least squares) of these points. 



111. As the origin of co-ordinates in the last article is arbi- 

 trary, the equations (i) evidently express the proposition that 

 in any system of parallel forces the sum of their moments about 

 any point is equal to the moment of their resultant about the same 

 point. In particular, the sum of the moments about any point on 

 the resultant is zero. 



This proposition may be regarded as a generalisation of the 

 principle of the lever referred to in Art. 106. It furnishes thei 

 convenient method of "taking moments" for the purpose of 

 determining the position of the resultant. 



112. Couple of Forces. The construction given in Art. 104 

 for the resultant of two parallel forces fails only when the two 



