MATHEMATICAL NOTES. 93 



that is, PM being parallel to EG, EM= \EB, and /. PB=PC, 

 or BG is bisected in the point of contact P. The same is true 

 of any other side, and therefore every side is bisected in the 

 point of contact. 



2. Let p, p, be two forces into which a given system on a 

 rigid body may be resolved, a, 6, their least distance, and in- 

 clination of their directions ; pp a sin 6 is invariable. (Senate- 

 House, 1833.) 



Let the line a meet the directions of p and p in P and P' 

 respectively. At P apply two forces equal and parallel to p', 

 and opposite each other. Thus the system of forces is replaced 

 by the couple pa, and by the force at P, which is the resultant 

 of p and p. Kesolve this, the resultant, along the axis of the 

 couple and in its plane. Then the former component can arise 

 only from the resolved part of p, as^>' is wholly in the plane of 

 the couple. Also, as the shortest distance is perpendicular to 

 both lines, it follows that the arm of the couple is perpendicular 

 at P to the plane which contains the two forces p and p. 

 Hence 6, their mutual inclination, is that of p on the plane of 

 the couple, and therefore p sin 6 is the part of the general re- 

 sultant resolved along the axis of the couple. Then, if the 

 general resultant makes an angle (f> with the axis, we have in 

 the usual notation 



pp a sin 6 = GR cos <f> = B.G cos </>. 



Now G cos <f>, as is known, or as may be easily shown, = G v , 

 the minimum maximorum moment of the system ; 



therefore ppa sin GJEl, 

 which is constant. 



