40J SCREW CO-ORDINATES. 37 



and four similar equations ; hence p n p n is proportional to the determinant 

 obtained by omitting the /t th column from the matrix or : 



i&amp;gt; 72. 7a. 74. 75&amp;gt; 7 



,, 0,, &,, S 4 , 8 3 , 8 8 



I 

 i, e,, e 3 , 6 4 , 6 5 , e 6 , | 



and affixing a proper sign. The ratios of p 1} ... p s , being thus found, the 

 actual values are given by 35. 



If there were a sixth screw f the evanescence of the determinant which 

 written in the usual notation is (a 1; /3 2 , 73, S 4 , e 5 , f 6 ) would express that the 

 six screws had a common reciprocal. This is an important case in view of 

 future developments. 



40. Co-ordinates of a Screw on a Cylindroid. 



We may define the screw 8 on the cylindroid by the angle I, which it makes 

 with a, one of the two principal screws a and /3. Since a Avrcnch of unit 

 intensity on 6 has components of intensities cos I and sin I on a and j3 ( 14), 

 and since each of these components may be resolved into six wrenches on 

 any six co-reciprocal screws, we must have ( 34) 



6 n = On cos I + fin sin I. 

 From this expression we can find the pitch of : for we have 



p e = Spj (: cos I 4- & sin If, 



whence expanding and observing that as a and /3 are reciprocal p 1 a. 1 @ 1 0, 

 and also that S,p 1 a l ~ = p a and f!h& &amp;gt;a *Pjh we have the expression already 

 given ( 18), viz. 



pe = p* cos 2 I+PP sin 2 1. 



If two screws, and &amp;lt;, upon the cylindroid, are reciprocal, then (m being 

 the defining angle of &amp;lt;), 



2p t (! cos I + Pi sin 1) (! cos m + {3 l sin m) = 0, 

 or p a cos I cos m+pft sin I sin m = 0. 



Comparing this with 20, we have the following useful theorem : 



Any two reciprocal screws on a cylindroid are parallel to conjugate 

 diameters of the pitch conic. 



Since the sum of the squares of two conjugate diameters in an ellipse is 

 constant, we obtain the important result that the sum of the reciprocals of the 

 pitches of two reciprocal screws on a cylindroid is constant *. 

 * Compare Octonions, p. 190, by Alex. M u Aulay, 1898. 



