The operation (c) of taking the complement of an angle (α) is of order (period) two, since 90° –(90° –α) = α. Similarly, the operation (s) of taking the supplement of an angle is of order two. If we perform these two operations in succession there results an operation (cs), which is of order four since it changes α into α + 90° Hence the group {c, s} generated by c and s is the dihedral rotation group of order 8. This fact follows directly from the general theorem, Two operations of order two generate the dihedral rotation group whose order is twice the order of their product.

The following system of algebraic inequalities is of great value and of wide application:

for all positive proper fractional values of a,

for positive proper fractional values of α1, α2, αn, and value of k Such that 2k<n; ∑r denoting the sum of the r-products of the α’s.

To find the position of equilibrium of a particle which is acted on by forces of given magnitudes directed towards given points.

This problem was suggested by Mr. P A. Hillhouse, and arose in the first instance from the consideration of forces acting in a crane. The following note does not offer a solution of the problem, but contains a proof by Mr. A. L. Jones that there is at most one position of equilibrium when the forces are all of one sign, i.e. either all tensions or all thrusts. This is followed by an investigation of the total number of solutions, real and imaginary, for all combinations of thrusts and tensions, when the magnitude only of each force, and not its sign, is given. This number is surprisingly large. It would of course be of more practical interest to find the number of real solutions for each particular combination of thrusts and tensions, but I have only succeeded in doing this when the given points are in a straight line, and in one or two other special cases. The results for these are stated without proof at the end of the note.