A Treatise on the Theory of Screws

306 THE THEORY OF SCREWS. [286- a screw a. But the initial velocity of the body in this case will not generally be cos (ar/) -r pa. It may be easily shown to be But we have also Q— ________ P tp^ß^ ’ whence in all cases ß™ßn = å-sraf. This formula is therefore much more general besides being more concise than that of § 281. 287. System with Two Degrees of Freedom. Let A, B, G, X, &c., and A', B', G', X', &c., be two homographic systems of points on a circle. These correspond respectively to two homographic systems of screws on the cylindroid according to the method of representation in Chap. XII. Then it is known, from geometrical principles, that if any two pairs, such as A, A' and B, B', be taken, the lines AB, BA intersect on a definite straight line, which is the axis of the homography. In general this axis may occupy any position whatever; if, however, it should pass through 0, the pole of the axis of pitch, then the homography will assume a special type which it is our object to investigate. In the first place, we may notice that under these circumstances the homography possesses the following characteristic:— Let A, B be two screws, and A’, B' their two correspondents; then, if A be reciprocal to B', B must be reciprocal to A'. For in this case AB' must pass through 0, and therefore BA' must pass through 0 also, i.e. B and A' must be reciprocal. This cross relation suggests a name for the particular species of homo- graphy now before us. The form of the letter indicates so naturally the kind of relation, that I have found it convenient to refer to this type of homography as Ghiastic. No doubt, in the present illustration I am only discussing the case of two degrees of freedom, but we shall presently see that chiastic homography is significant throughout the whole theory. 288. A Geometrical Proof. It is known that in the circular representation the virtual coefficient of two screws is proportional to the perpendicular distance of their chord from the pole of the axis of pitch (§ 61).