440 THE THEORY OF SCREWS. [400, 



The intervene itself is F (&), where F expresses some function; and, 

 accordingly, 



When we substitute in this expression the value of 6 given above, we have 

 an identity which is quite independent of the particular 8. We must, there 

 fore, determine the functions so that this equation shall remain true for all 

 values of X, and all values of p. The formulae must therefore be true when 

 differentiated 



d6_ __(/*-X )OA-X&quot;) d6 = (X-X )(X&quot;-X) 



dx t-x) 2 d^ (/i-x) 2 



whence, 



or (X - X )(X - X&quot;) &amp;lt; (X) = O - X )(/A - X&quot;) f (/*), 



which has the form 



Considering the complete independence of both X and /JL, this equation re 

 quires that each of its members be independent alike of X and /*. We shall 

 denote them by H (X X&quot;) where H is a constant, whence 



(X - X )(X - X&quot;) (j&amp;gt; (\) = H(\ r - X&quot;), 





JUV 



-v x-x V 



whence, integrating and denoting the arbitrary constant by C, 



(f&amp;gt;(\)=H [log (X - X ) - log (X - X&quot;)] + G ; 

 similarly, 



and, finally, we have for the intervene, or (X) &amp;lt;/&amp;gt; (^), the expression, 



This expression discloses the intervene as the logarithm of a certain an- 

 harmonic ratio. 



We may here note how a difficulty must be removed which is very 

 likely to occur to one who is approaching the non-Euclidian geometry for the 



