284 NOTE ON POINSOT’S MEMOIR ON ROTATION. 
trical manner. In our modern text books the analytical method is 
mainly adopted, and it has seemed to me that the beauty and sim- 
plicity of the system have thereby been much overlooked. In fol- 
lowing, and possibly simplifying by a more elementary geometry, 
Poinsot’s course, we commence with the general reduction of a set 
of statical forces to a single resultant force and a single resultant 
couple. 
1. Let P be one of a set of forces acting at assigned points of a 
rigid system, and let A be a point arbitrarily assumed as an origin. 
At A apply two opposite forces, each equal and parallel to P. Then 
the original force P is replaced by an equal and parallel force acting 
at A, and acouple. Each of the forces of the system may be treated 
in the same way, and the whole set will be replaced by a set of forces 
acting at A, (which may be combined into a single Resultant &), 
and a set of couples which may be combined into a single couple @. 
2. Since RF is compounded of a set of forces which are severally equal 
and parallel to those of the original set, & evidently remains the 
same in direction and magnitude, whatever origin be assumed; Gin 
general varies for different origins in both respects, but evidenjly 
remains the same for all origins which lie in the direction of &. 
3. To examine the changes which G undergoes in passing from 
one origin to another, let B be any other origin, and at B apply two 
opposite forces, each equal and parallel to R. We have then, & at 
B, the couple O, and the newly introduced couple Ra (a being the 
distance between the directions of R at A and R# at B). Now 
suppose G to be resolved into two couples, whose axes are severally 
parallel and perpendicular to R; these will be, G cos0, and G sin@, 
where @ is the angle between A and the axis of G. Then the axis 
of the couple Ra being perpendicular to R, this couple will combine 
with G sin@, but will not affect the other resolved part G cos@. 
Hence, whatever origin be adopted, the resolved part G cos0, whose 
axis is in direction of the resultant force, always remains the same. 
The other component of the couple admits of all values according to 
the origin adopted. We may therefore adopt an origin (or in fact 
a line of origins parallel to R) such that this other component shall 
be zero, and we have then remaining a couple whose axis is in the 
direction of the resultant force. In this case, the resultant couple 
evidently has its least possible value. 
4. Calling G’ this value of it, on transferring to another origin 
as in (3), the new couple will be compounded of G’ and fa, the 
axes of which are at right angles to each other; and the new couple 
