205] PLANE REPRESENTATION OF THE THIRD ORDER. 205 



cosines are proportional to 8 } , # 2 &amp;gt; ;1 ; for if we take the point infinitely 

 distant we find that the equations reduce to 



x _ y _ z 

 6 l &, $3 



Accordingly, the line drawn parallel to the screw through the origin has its 

 direction cosines proportional to 0,, 2 , 3 , and hence the actual direction 

 cosines are 



ft ft 3 



The cosine of the angle between two screws, 6 and &amp;lt;/&amp;gt;, will therefore be 



By the aid of the conic of infinite pitch we can give to this a geometrical 

 interpretation. 



The co-ordinates of a screw on the straight line joining 6 and &amp;lt;f&amp;gt; will be 



ft + X&amp;lt;j , # 2 + ^-$2&amp;gt; ft + X&amp;lt; 3 . 



If we substitute this in the equation to the conic of infinite pitch we obtain 



ft 2 + ft 2 + 9* -H 2X (#!&amp;lt;/&amp;gt;, + 2 &amp;lt;/&amp;gt; 2 + 3 3 ) + \- (fa- 4- &amp;lt;/&amp;gt;.;- + &amp;lt;/) = 0. 

 Writing this in the form 



aX 2 + 2&X + c = 0, 



of which Xj and X 2 are the roots, we have, as the four values of X, corre 

 sponding, respectively, to the points 6 and &amp;lt;/&amp;gt;, and to the points in which 

 their chord cuts the conic of infinite pitch, 



X 1( Xo, 0, oo. 



The anharmonic ratio is 



^ 

 V 

 or 



b - V6 2 - ac 

 b + V6 2 - ac 



If to be the angle between the two screws, 6 and &amp;lt;, then 



b 



COS G) = , 



Vac 

 and the anharmonic ratio reduces to 



an*-, 

 whence we deduce the following theorem : 



