214] FREEDOM OF THE FOURTH ORDER. 223 



is of course 



(Pe + P*) cos W) - sin (0&amp;lt;f&amp;gt;) d^ = 2p e cos (0&amp;lt;j&amp;gt;), 



or 



(P&amp;lt;t&amp;gt; ~ Pe) cos (0&amp;lt;f&amp;gt;) - sin (0&amp;lt;f&amp;gt;) d 64&amp;gt; = 0. 



If then p^ = p 6 we must have sin (#(/&amp;gt;) c^ = 0, which requires that and 

 must intersect at either a finite or an infinite distance. 



In the case where &amp;lt; is at right angles to 6 it follows from the formula 

 ^ob cos (0&amp;lt;fi)p 6 that CT04, = 0, or that and are reciprocal. But two screws 

 which are at right angles and also reciprocal must intersect, and hence we 

 have the following theorem. 



If 6 be a screw of stationary pitch in an n-system, then any other screw 

 belonging to the n-system and at right angles to Q must intersect 6. 



If (/&amp;gt; belongs to an w-system its co-ordinates must, on that account, satisfy 

 6 n linear equations. If it be further assumed that &amp;lt; has to be perpen 

 dicular to 0, then the co-ordinates of &amp;lt; have to satisfy yet one more equa 

 tion, i.e. 



. dR , dR 



ft&amp;gt;:-*;BETft 



In this case &amp;lt;/&amp;gt; is subjected to 7 n linear equations. It follows ( 76) that 

 (f&amp;gt; will have as its locus a certain (n l)-system, whence we have the following 

 general theorem. 



If 6 is a screw of stationary pitch in an n-system P then among the 

 (n Y)-sy stems included in P there is one Q such that every screw of Q intersects 

 at right angles. 



These theorems can also be proved by geometrical considerations. If a 

 screw 6 have stationary pitch in an w-system it follows a fortiori that 6 must 

 have stationary pitch on any cylindroid through 6 and belonging wholly to 

 the n-system. This means that must be one of the two principal screws 

 on such a cylindroid. Choose any other screw &amp;lt; of the system and draw the 

 cylindroid (0, &amp;lt;/&amp;gt;) then 6 is a principal screw, and if 6 and the other principal 

 screw on the cylindroid be two of the co-reciprocal screws of reference, then 

 the co-ordinate of &amp;lt; with respect to is cos(0&amp;lt;) ( 40). But that co-ordinate 

 must also have the general form -nr^ -rp l , whence at once we obtain 



Let be a screw of stationary pitch in a three-system, and let &amp;lt;/&amp;gt; and ^r 

 be any two other screws in that system. Then 6 is one of the principal 

 screws on the cylindroid (#&amp;lt;/&amp;gt;) ; let &amp;lt;j be the other principal screw on that 



