Forces not parallel acting in different Planes, 41 



Now this expression for the distance FE, is precisely that 

 which we should have found for the distance at which the 

 resultant passes, if the three forces p, </, r, had all been in the 

 plane ZF, and had passed through the points A', C', .B', corres- 

 ponding to the points A, C, , through which they actually pass. 

 If, therefore, we imagine the straight line FX perpendicular to 

 the plane ZF, we shall find the distance FE, of the resultant 

 from this straight line, by taking the sum of the moments with 

 reference to this line (as if the forces, retaining their distances 

 respectively from this line, were all in the plane ZF, to which 

 this line is perpendicular), and dividing this sum of the moments* 

 by the sum of the forces. 



To determine the point , therefore, it only remains to find 

 the distance EE, or (by taking EE" parallel to ZF) the distance 

 FE') at which this same force passes, from ZF. Now it is mani- 

 fest from what we have said with respect to the distance FEf, 

 that in order to find the distance FE", we have only to imagine 

 a plane passing through ZF, perpendicular to the direction of the 

 forces, and then to take the sum of the moments with respect to 

 ZF (the intersection of this plane with the plane ZF), as if the 

 forces, without changing their distances respectively from the 

 plane ZF, were all in the plane XV, and to divide this sum of the 

 moments by the sum of the forces. We should then have every 

 thing which is necessary for fixing the point E, through which 

 the resultant passes. , Top. i. 



Of Forces the Directions of which are neither in the same Plane nor 

 parallel to each other. 



71. Letp, <?, r, be three forces directed in the manner repre- F1 24 

 sented in the figure, and situated in three different planes. Sup- 

 pose any plane XZ meeting in H the direction of p, in F the 



* It must he observed, once for all, that by the general term, sum 

 of the moments, is to be understood the sum of the moments of the 

 forces that tend to turn the system in one direction, minus the sum 

 of those which tend to turn it in the opposite direction. By sum of 

 the forces, also, is to be understood the sum of those which act in one 

 direction, minus the sum of those which act in the opposite direction. 



Mech. 6 



