Parallel Forces acting in different Planes. 39 



to FC' and FB", that is, for the resultant of all the proposed 

 forces. 



Of parallel Forces which act in different Planes. 



70. Let there be three forces/*, q, r, directed according to Fig 23. 

 the lines Ap, B q, C r, parallel to each other, but situated in 

 different planes. 



Imagine a plane XZ to which the three straight lines Ap, &c., 

 are perpendicular, and another plane ZV to which they are par- 

 allel, and let A, B, C, be the three points where these lines meet 

 the plane XZ. 



The two forces j9, r, are in the same plane, the intersection of 

 which with the plane XZ is the straight line AC. These two335,324> 

 forces may therefore be reduced to a single one p', equal to 50> 

 p :'- r, having a direction parallel to that of the components, and 67 

 passing through a point D, such that;) X AD r x CD. 



The two forces p', q, are in the same plane, the intersection 

 of which with the plane XZ is BD. These may accordingly be 

 reduced to a single one p, equal to p' -f q, that is, equal to 



P -f q -f- r, 



having a direction parallel to that of p' and r, and passing through 

 a point E, such that p' x DE = q X BE. -It follows, therefore, 

 from this and what is said above, that any number of forces, the 

 directions of which are parallel, may be reduced to a single one, equal 

 to the sum of those which act in one direction, minus the sum of those 

 which act in the contrary direction, whether the given forces are in 

 the same or in different planes. 



Let us now inquire more particularly how we are to deter- 

 mine through what point the resultant passes. 



If from the points A, D, C, B, E, we draw the lines AA', 

 DD, CC, BE', EE 1 , perpendicularly to the common intersection 

 of the two planes XZ, ZV -, on account of the parallels A A, DD, f 

 CC*, we shall have 



AD : CD : : AD' : C'D\ 



Now the equation p X AD = r X CD, found above, gives 

 AD : CD : : r : p ; 



