444 THE THEORY OF SCREWS. [403, 



The intervene between the objects X and X is 



iHir. 



Nor is this inconsistent with the fact that \ = zero denotes two coincident 

 objects, as does also X= infinity. In each of these cases the coincident 

 objects are at infinity, and the intervene between two objects which coalesce 

 into one of the objects at infinity has an indeterminate value, and may thus, 

 of course, be iH-rr, as well as anything else. 



404. Intervenes on Different Ranges in a Content. 



Let us suppose any two ranges whatever. There are an infinite number 

 of objects on one range, and an infinite number on the other. The well- 

 known analogies of homographic systems on rays in space lead us to inquire 

 whether the several objects on the two ranges can be correlated homo- 

 graphically. Each object in either system is to correspond definitely with 

 a single object in the other system. 



We determine an object on a range by its appropriate X. Let the 

 corresponding object on the other range be defined by X . The necessary 

 conditions of homography demand that for each X there shall be one X , and 

 vice versa. Compliance with this is assured when X and X are related by 

 an equation of the form 



PXX + Q\ + R\ + 8 = 0. 



Let \i, X 2 , X 3 , X 4 be any four values of X, and let X/&amp;gt; X,/, X*/, X/ be the cor 

 responding four values of X , then, by substitution in the equation just 

 written, and elimination of P, Q, R, S, it follows that 



Xi Xs Xj X 4 A! Xs A-2 X4 



X^ \s Xj X 4 X 2 X 3 Xj X 4 



We now introduce the following important definition : 



By the expression, anharmonic ratio of four objects on a range, is meant 



the anharmonic ratio of the four values of the numerical parameter by which 



the objects are indicated. 



We are thus enabled to enunciate the following theorem : 



When the objects on two ranges are ordered homographically, the an 



harmonic ratio of any four objects on one range equals the anharmonic ratio 



of their four correspondents on the other. 



Three pairs of correspondents can be chosen arbitrarily, and then the 

 equation last given will indicate the relation between every other X and its 

 corresponding X . 



Among the different homographic systems there is one of special im 

 portance. It is that in which the intervene between any two objects in 



