400] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 439 



whence we deduce 



X &amp;lt; (&amp;gt;) = /*&amp;lt;/&amp;gt; (/A 



As X and p are perfectly independent, this equation can only subsist by 

 assuming for &amp;lt; a form such that 



\&amp;lt;j&amp;gt; (\) = H, 

 where H is independent of X. Whence we obtain 



&amp;lt;/&amp;gt; = f, 



and, by integration, 



(X) = H log X + constant. 



The intervene is now readily determined, for 



&amp;lt;j&amp;gt; (X) - &amp;lt;/&amp;gt; O) = H log X - H log p, = H log - . 



f 



We therefore obtain the following important theorem which is the well- 

 known* basis of the mensuration of non-Euclidian space : 



Let #1, # 8 , #g, # 4 , and y lt y a , y s , y* be the two objects at infinity on a range, 

 and let a^ + X^i, # 2 + Xy 2 , x 3 + \y 3 , x + ^y, and x l + /u# 1} x 2 + /j,y 2 , oc 3 + ^y 3 , 

 i + py* be any two other objects on the range, then their intervene will be 

 expressed by 



H (log X - log fi) t 



where H is a constant depending upon the adopted units of measurement. 



It will be useful to obtain the expression for the intervene in a rather 

 more general manner by taking the equation in X and /A, for objects at the 

 intervene 8, as 



Let X and X&quot; be the two roots of this equation when //. is made equal to X. 

 It follows that what we have just written may be expressed thus: 



X^ + X (0 - IX - IX&quot;) + fi (- 6 - ^X - IX&quot;) + X X&quot; = 0. 



For, if X = p, this is satisfied by either X or X&quot;, while 6 disappears. 6 is, of 

 course, a function of the intervene, and it is only through 6 that the inter 

 vene comes into the equation. By solving for 9, we find 



* Professor George Bruce Halsted remarks in Science, N. S., Vol. x., No. 251, pages 545557, 

 October 20, 1899, that &quot;Koberto Bonola has just given in the Bolletirw di Bibliografia e Storia 

 della Scienze Matematiche (1899) au exceedingly rich and valuable Bibliografia sui Fondamenti, 

 della Geometria in relazione alia Geometria non-Euclidea in which he gives 353 titles.&quot; 



