NOTE ON POINSOT’S MEMOIR ON ROTATION. 285 
will therefore have the same value so long as a remains the same— 
that is:—for all origins lying on a circular right cylinder about the 
line of origins spoken of, and for this reason this line of origins is 
ealled by Poinsot the central azis, 
5. Since # is the same for all origins, the set of forces is not 
reducible to a single couple, unless it should happen that kK — 0. 
In this case, the forces must be capable of being represented in 
magnitude and direction by the sides of a polygon (or of several 
polygons) taken in order. If the forces were represented in position 
also by the sides of the polygon, and the polygon moreover were a 
plane one, then the magnitude of the resultant couple would be 
independent of the position of the forces with regard to the system, 
being in fact represented geometrically by the area of the polygon. 
6. Since G cos0 is the same for all origins, the set of forces is not 
reducible to a single force, unless it should happen that G cos 02—0. 
That this may be the case, we must either have G — 0, or 
0 =53 that is:—we must find at our assumed origin either the 
resultant couple vanishing, or else its axis at right angles to the 
direction of the resultant force. If the latter be the case at any 
one origin, it must plainly be so at all origins, and it is easy to see 
in what way the reduction to a single force is effected. For the 
plane of the couple can be moved so as to contain R, the couple can 
be turned till one of its forces is opposite to R, and the arm can be 
altered till this force is equal to &; these two forces being then 
removed, there remains the other force (/) of the couple for the 
single resultant, acting in a line whose distance from the direction 
of & through our assumed origin is equal to = (Of course if R 
should happen to be O, this transformation is illusory.) This con- 
dition is evidently satisfied when the forces of the system are all 
parallel, and the single resultant in this case is equal to the algebraic 
sum of the forces, provided that this sum be finite. 
7. Any set of forces can also in the general case be reduced, 
in an infinite variety of ways, to two, acting along lines which 
neither meet nor are parallel. For, let the couple G be trans- 
ferred till the direction of one of its forces intersects that of R; 
then these two can be compounded into a single force, and this and 
the remaining force of the couple constitute the two forces acting 
ag stated. The elements of these two forces are of course not 
