280 THE THEORY OF SCREWS. [266, 



Let the angle AOh be denoted by A, the angle ^40?; by B, and the angle 

 A OX by &amp;lt;f). To find the component \ we must decompose A, , a twist on 

 X, into two components, one on 77, the other on the first screw of reference. 

 The component on 77 can be resolved along the other five screws of reference, 

 since the six form one system with a common reciprocal. If we denote by 

 77 the component on 77, we then have 



X \! 77 



sin (B - A) = sin (&amp;lt; - B) = sin(&amp;lt;/&amp;gt;-J) 



and if a and 6 be the pitches of the two principal screws on the cylindroid, 

 we have for the pitch of X the equation 



p = a cos 2 &amp;lt;/&amp;gt; + b sin 2 &amp;lt;f&amp;gt; ; 



also -3^- = ~ V , because the effect of a change in X, is to move the screw 

 aXj d(f&amp;gt; ctXj 



along this cylindroid. 



Air u , sin (0 - .B) 



We have Xj = 77 -r } --- A , 



sm (0- A) 



and as the other co-ordinates are to be left unchanged, it is necessary that 

 77 be constant, so that 



d\ _ ,sm(B A) 

 ~d$~ rj sin^(&amp;lt;f&amp;gt;-A) 



dp . . sin 2 (&amp;lt;4 A) 



and hence ,^- = (6 - a) sin 2&amp;lt;f&amp;gt; , . ^ - 4r . 



ctXj ^T)sm(B-A) 



A , d\ d\ dd&amp;gt; 



Also = ^_ = _ cos ( ( f ) _^) j 



aXj rf0 rfXj 

 Hence, substituting in the equation 



we deduce a = b tan &amp;lt;/&amp;gt; tan A : 



but this is the condition that X and the first screw of reference shall be 

 reciprocal ( 40). 



267. Analogy to Orthogonal Transformation. 



The emanants of the second degree are represented by the equation 



when F is the function into which / becomes transformed when the co 

 ordinates are changed from one set of screws of reference to another. If 

 we take for / either of the functions already considered, these equations 



