222 THE THEORY OF SCREWS. [214 



add the products to the former equation. We can then equate the co 

 efficients of 80!,... 8# 6 severally to zero, thus obtaining 



J~D 



2p 6 6 - ja Pe + *A + A^e = 0. 

 av 6 



Choose next from the four-system any screw whatever of which the co 

 ordinates are $ 15 ... &amp;lt;. Multiply the first of the above six equations by &amp;lt;j&amp;gt; 1} 

 the second by &amp;lt; 2 , &c. and add the six products. The coefficients of X and //, 

 vanish, and we obtain 



dR , dR 



The coefficient of p g is however merely double the cosine of the angle 

 between 6 and &amp;lt;. This is obvious by employing canonical co-reciprocals 

 in which 



.R = (0i + 0*y + (0, + e t y + (0, + 0,y, 



whence 



dR dR 



(0, + 2 )+2 (0 3 + 4 ) (0 3 + t ) + (0 8 + 00 (0, + 6 ) = 2 cos 



We thus obtain the following theorem, which must obviously be true for 

 other values of n besides four. 



If (f) be any screw of an n-system and if 6 be a screw of stationary pitch 

 in the same system then -ar^ = cos (6(f&amp;gt;)p e . 



Suppose that there were two screws of stationary pitch and in an ??- 

 system. Then 



13- = cos 



If p e and PCJ, are different these equations require that 



BJ^ = ; cos (#&amp;lt;/&amp;gt;) = ; 



i.e. the screws are both reciprocal and rectangular and must therefore 

 intersect. 



We have thus shown that if there are two stationary screws of different 

 pitches in any w-system, then these screws must intersect at right angles. 



In general we learn that if any screw of an ?i-system has a pitch equal 

 to that of a screw of stationary pitch in the same system, then and 

 must intersect. For the general condition 



* = cos (#&amp;lt;/&amp;gt;) p 6 



