9 i.] INTRODUCTION. 49 



metric sum of its components, it follows that the projection of 

 the resultant on any line equals the algebraic sum of the pro- 

 jections of its components. This proposition is sometimes 

 expressed in the following form : the resolved part of the 

 resultant in any direction is equal to the algebraic sum of 

 the resolved parts of the components. ^ 



Let / be the line on which we project (Fig. 17), and let (/, R] 

 (/,'P), (l/Q) denote the angles it makes with the resultant R 

 and the components P, Q, respectively ; then 



R cos (/, ( R)=P cos (/, P) + Q cos (/, ' Q). 



90. Varignon's Theorem. Multiplying the last equation by any 

 length OS=s taken through the initial point O of R and at 

 right angles to /, we obtain 



R-scos(t, R) = P-scos(l, P) + Q-scos(t, Q\ 



or since s cos (/, R} r, scos (/, P) =/, 

 s cos (/, Q) = q, where r, p, q are the 

 perpendiculars let fall from 5, on 

 R, P, Q, respectively, 



In this form the proposition is in- 



dependent of the direction of the 



line / and holds for any point 5 in 



the plane of the parallelogram. Fig. 17 



91. Moment of a Force. The product of a force into its per- 

 pendicular distance from a point is called the moment of the 

 force about the point. It is taken with the positive or negative 

 sign according as the force as seen from the point is directed 

 counter-clockwise or clockwise. 



The proposition of Art. 90, Pp + Qq = Rr, can now be stated 

 in the following form : the algebraic sum of the moments of any 

 tivo intersecting forces about any point in their plane is equal to 

 the moment of their resultant about the same point. 



PART II 4 





