206 THE THEORY OF SCREWS. [205- 



The angle between two screws is equal to \i times the logarithm of the an- 

 ha,rmonic ratio in which their corresponding chord is divided by the infinite 

 pitch conic. 



The reader will be here reminded of the geometry of non-Euclidian 

 space, in which a magnitude, which in Chapter XXVI. is called the Intervene, 

 analogous to the distance between two points, is equal to ^i times the 

 logarithm of the anharmonic ratio in which their chord is divided by the 

 absolute. We have only to call the conic of infinite pitch the absolute, and 

 the angle between two screws is the intervene between their corresponding 

 points. 



206. Screws at Right Angles. 



If two screws, 9 and &amp;lt;/&amp;gt;, be at right angles, then 



$]&amp;lt;l + 0. 2 &amp;lt;f).2 + 0;,,(f&amp;gt; 3 = 0. 



In other words, 6 and &amp;lt;/&amp;gt; are conjugate points of the conic of infinite pitch, 



V + $+ &- & 



All the screws at right angles to a given screw lie on the polar of the point 

 with regard to the conic of infinite pitch. Hence we see that all the screws 

 perpendicular to a given screw lie on a cylindroid. This is otherwise obvious, 

 for a screw can always be found with an axis parallel to a given direction. 

 If, therefore, a cylindroid of the system be taken, a screw of the system 

 parallel to the nodal axis of that cylindroid can also be found, and thus we 

 have the cylindroid and the screw, which stand in the relation of the pole and 

 the polar to the conic of infinite pitch. 



A point on the conic of infinite pitch must represent a screw at right angles 

 to itself. Every straight line cuts the conic of infinite pitch in two points, and 

 thus every cylindroid has two screws of infinite pitch, and each of these 

 screws is at right angles to itself. 



In general, the direction cosines of the nodal axis of a cylindroid are 

 proportional to the co-ordinates of the pole of the line corresponding to the 

 cylindroid with respect to the conic of infinite pitch. 



207. Reciprocal Screws. 



If lt 0,, 3 be the co-ordinates of a screw, and n &amp;lt;/&amp;gt;,, &amp;lt;/&amp;gt; 3 those of another 

 screw, then it is known, 37, that the condition for these two screws to be 

 reciprocal is 



We are thus led to the following theorem, which is of fundamental importance 

 in the present investigation : 



