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 H l 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 



*t 



derived from making it equal to -= , for then the entire circuit of any range 



z 



ft 



is TT. We now stipulate that H is also to be ^ . The circuit of a star 



it 



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 # 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 4- 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 formulae 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. 



