A Treatise on the Theory of Screws

452 THE THEORY OF SCREWS. [412, vene are also objects of zero departure, and conversely. Thus we see that the two systems of critical objects in the extent coalesce into a single system in consequence of the assumption in Axiom xi. Each consecutive pair of critical objects determine a range, which is a range of infinite departure as well as of zero intervene. The expression infinite objects will then denote objects which possess the double property of having, in general, an infinite intervene from other objects in the extent, and of being also the vertices of stars of zero departure. The expression infinite ranges will denote ranges which possess the double property of having, in general, an infinite departure with all other ranges, and which consist of objects, the intervene between any pair of which is, in general, zero. There is still one more point to be decided. The measurement of depar- ture, like that of intervene, is expressed by the product of a numerical factor with the logarithm of an anharmonic ratio. This factor is H for the intervene. Let us call it H' for the departure. What is to be the relation between H and li"( Here the analogy of geometry is illusory; for, owing to the coincidence between the points of infinity on a straight line, H has to be made infinite in ordinary geometry, while H' must be finite. But in the present more general theory H is finite, and we have found much convenience derived from making it equal to — , for then the entire circuit of any range is 7t. We now stipulate that H' is also to be — . The circuit of a star will then be tt also. With this assumption the theory of the metrics of an extent admits of a remarkable development. Let x, y, z be any three objects. Let a, b, c denote the intervenes between y and z, z and x, x and y, respectively. Let the departure between the ranges from x to y and x to z be denoted by A, from y to z and y to x be denoted by B, and from z to x and z to y be denoted by C. Then, sin A __ sin B sin G sin a sin b sin c ’ cos a = cos b cos c + sin b sin. c cos A, cos b = cos c cos a + sin c sin a cos B, cos c = cos a cos b + sin a sin b cos C. Thus the formulæ of spherical trigonometry are generally applicable through- out the extent*. I learned this astonishing theorem from Professor Heath’s very interesting paper Phil Trans. Part n. 1884.