456 THE THEORY OF SCREWS. [416, 



content, conducted subject to the condition that the intervene between 

 every pair of objects shall equal that between their correspondents. 



417. On the Character of a Homographic Transformation 

 which Conserves Intervene. 



In all investigations of this nature the behaviour of the infinite objects 

 is especially instructive. In the present case it is easily shown that every 

 object infinite before the transformation must be infinite afterwards 

 when moved to . For, let X be any object which is not infinite before the 

 transformation, nor afterwards, when it becomes X . Then, by hypothesis, 

 the intervene OX is equal to O X ; but OX is infinite, therefore O X must 

 be also infinite, so that either or X is infinite ; but, by hypothesis, X is 

 finite, therefore must be infinite, so that in a homographic transformation 

 which conserves intervene, each object infinite before the transformation remains 

 infinite afterwards. 



It follows that in the space representation each point, representing 

 an infinite object, and therefore lying on the infinite quadric U=Q must, 

 after transformation, be moved to a position which will also lie on the 

 infinite quadric. Hence we obtain the following important result : 



In the space representation of a homographic transformation which con 

 serves intervene, the infinite quadric U=Q is merely displaced on itself. 



A homographic transformation of the points in space will not, in general, 

 permit any quadric to remain unchanged. A certain specialization of the 

 constants will be necessary. They must, in fact, satisfy a single condition, 

 for which we shall presently find the expression. 



Let cc lt oc 2 , x 3 , # 4 be the quadriplanar co-ordinates of a point, and let us 

 transform these to a new tetrahedron of which the vertices shall have as 

 their co-ordinates with respect to the original tetrahedron 



1 . rr /y&amp;gt; 



&quot;I ) *2 ) X 3 &amp;gt; &quot; 4 &amp;gt; 



v&quot; &amp;lt;r&quot; r&quot; &amp;lt;r&quot; 



*Tl i a 2 ^3 &amp;gt; &quot;^l &amp;gt; 



If then X l} X 2 , X 3 , X 4 be the four co-ordinates of the point referred to 

 the new tetrahedron 



+ Xi X* + x^&quot;X s + 



