Forces not parallel acting in different Planes. 43 



are reduced to two, which do not admit of being represented by 

 one, except in the case where the resultant of the forces which 

 act in the plane XZ, happens to meet the resultant of the forces 

 perpendicular to this plane. 



73. We can accordingly, in the manner above explained, find 

 the two resultants of any number of forces directed in different 

 planes. But, although the method here pursued may be useful 

 in certain cases, there are many in which it is not the most con- 

 venient. We proceed, therefore, to make known another. 







Let p be any one of the proposed forces, and AE the line which Fig. 25. 

 represents it. From any fixed point X draw the three straight 

 lines XZ, XY, XT, perpendicular each to the plane of the other 

 two. These mutually perpendicular lines are called rectangular 

 co-ordinates. If now, upon AE as a diagonal, we form the rectan- 

 gular parallelogram ADBC, having its plane perpendicular to the 

 plane YXT, and its side BC parallel to XZ ; and then upon BD 

 as a diagonal we form the rectangular parallelogram DFBE, 

 having its plane parallel to the plane YXT, and its sides BF, BE, 

 parallel to the straight lines XT, XY, respectively ; it is evident 

 1. That for the force AE we may substitute the force BC parallel 

 to XZ, or perpendicular to the plane YX T, and the force BD par- 

 allel to this latter plane; 2. That for the force BD we may 

 substitute the force BE parallel to XY, or perpendicular to the 

 plane ZXT, and the force BF parallel to XT, or perpendicular 

 to the plane ZXY-, so that the force p, or AB, is decomposed into 

 three forces parallel to three rectangular co-ordinates, or (which 

 is the same thing) into three forces perpendicular to three mutu- 

 ally perpendicular planes. 



Now what has been said of the force p is evidently applicable 

 to any other force not perpendicular to one of the three planes. 

 If therefore all the forces like p, be considered as thus decompos- 

 ed 5 and we afterwards reduce to a single force all the forces per- 

 pendicular to the plane ZXT, the same thing being done with re- 6 

 spect to all the forces perpendicular to the plane ZXY, and also 

 with respect to all the forces perpendicular to YXT, it will be seen 

 that we may reduce any number of forces, directed in different 

 planes, to three forces perpendicular to three planes, these planes 

 being perpendicular respectively to each other. 



