42 Statics. 



direction of q, and in L the direction of r. As a force may be con- 

 sidered as applied at any point whatever in its direction, let us 

 suppose the three forces to be applied at the points H, F, L, and 

 to be represented by the lines HV, FT, LK, prolongations respec- 

 tively of the directions of the several forces below the plane XZ. 

 Let us also imagine planes passing through the lines All, BF, 

 CL, perpendicularly to the plane XZ, the intersections with XZ 

 being represented by the straight lines GHN, EFY, DLM. This 

 premised, it is evident that we may decompose each of the forces 

 in question into two others, one of which shall be in the plane 

 XZ, and the other perpendicular to this plane. We can, for ex- 

 ample, decompose the force H V into two others, one directed ac- 

 cording to HN, and the other according to HO, perpendicular 

 to the plane XZ; so that for the three forces HV, FT, LK, may 

 be substituted the six forces 



HN, FY, LM, HO, FS, LI, 



the three first of which are in the plane XZ, and the three last 

 perpendicular to this plane. 



Now the three forces HN, FY, LM, may be reduced to a 

 69 - single one, which shall also be in the plane XZ; and the three HO, 

 FS, LI, may likewise be reduced to a single one, which shall be 

 70. perpendicular to the plane XZ. Accordingly, whatever be the num- 

 ber of given forces, and whatever their directions, we may always 

 reduce them to two at the most, one being in the direction of a known 

 plane, and the other perpendicular to this plane. 



Although the demonstration of this proposition may appear 

 to be adapted only to those cases where all the forces meet the 

 plane XZ, it will be seen, with a little attention, that it has a gen- 

 eral application. For, after having reduced all the forces that 

 meet this assumed plane to two, we may conceive this plane, 

 without ceasing to meet these two resultants, so placed as to 

 coincide with the directions of those that were at first parallel 

 to it ; and the given forces must be either parallel to the assum- 

 ed plane, or such as being produced will meet it. 



72. With respect, therefore, to forces exerted in different 



planes, the result is not the same as that with respect to forces 



whose directions are in the same plane. The latter forces, as we 



?0, have seen, may always be reduced to a single one. The former 



