138] PLANE REPRESENTATION OF DYNAMICAL PROBLEMS. 127
these components must, when compounded, produce the twist velocity w about
X', and, accordingly, we have
BX . B’X' öAX A’X'
a AB~W A'B7’ ^AB~WA’B"
Retaining A, B, A', B', as before, let us now introduce a second pair of points,
T and Y’, instead of X and X', and writing m instead of w, we have
■ BY .,B'Y' 6AY ,,A'Y'
a AB~a A'B' ’ ß AB ” A'B' ’
whence, eliminating d, ß, å>, &>', we have
BX BY B’X’ B'Y'
AX ' AY’’ AX' ' A'Y"
As the length of a chord is proportional to the sine of the subtended
angle, we see that the anharmonic ratio of the pencil, subtended by the four
points A, B, X, F at a point on the circumference, is equal to that subtended
by their four correspondents, A', B', X', Y'. We thus learn the following
important theorem:—
A system of points on the representative circle, regarded as impulsive
screws, and the corresponding system of instantaneous screws, form two homo-
graphic systems.
138. The Homographic Axis.
Let A, B, C, D (fig. 21) represent four impulsive screws, and let A', B’,
C, D' be the four corresponding instantaneous screws. Then, by the well-
Fig. 21.
known homographic properties of the circle, the three points, L, M, N, will
be collinear, and we have the following theorem:—