314 



THE THEORY OF SCREWS. 



[295, 



The calculation* presents no difficulty and the result is as follows : 



= cos 6 cos (f&amp;gt; 



+ cos 6 sin (f&amp;gt; 



+ sin 6 cos &amp;lt;/&amp;gt; 



+ sin 9 sin 



+p a cos 



cos 



- -S7 a cos 



cos (a??) [r a , cos (a) *3- j cos (a?;)] 

 ***, [cos (a) cos (77) - cos (a?;) cos 



+p a cos 



r a , cos 



- OT a cos 



cos (a) [r a , cos (a) nr a f cos (CM?)] 

 -GTpt [cos (a) cos (/ify) cos (arj) cos 

 cos (at)) \ixfr cos (a) -BT^ cos (a?;)] 



+ p a COS (77) [tjp, cos 



a, [cos (a) cos (/ify) cos (atj) cos 



+ PP cos (a^) [or^, cos (a^) tsfc cos (a??)] 



cos / r p , cos - ^ j cos 

 _+ ^-B3-a| [cos (af ) cos (firf) cos (a??) cos 



296. An exceptional Case. 



A few remarks should be made on the failure of the correspondence 

 when the principal planes of the two cylindroids are at right angles ( 294). 

 It will be noted that though this equation suffers a slight reduction when 

 the principal planes of the two cylindroids are at right angles yet it does 

 not become evanescent or impossible. For any value of 6 defining a screw 

 on one cylindroid, the equation provides a value of &amp;lt; for the correspondent 

 on the other cylindroid. Thus we seem to meet with a contradiction, for 

 while the argument of 294 shows that in such a case the homography 

 is impossible, yet the homographic equation seemed to show that it was 

 possible and indeed fixed the pairs of correspondents with absolute 

 definiteness. 



It is certainly true that if two cylindroids A and P admit of the cor 

 relation of their screws into pairs whereof those on P are impulsive screws 

 and those on A are instantaneous screws, the pairs of screws by which the 

 homographic equation is satisfied will stand to each other in the desired 

 relation. If, however, the screws on two cylindroids be correlated into 

 pairs in accordance with the indications of the homographic equation, 

 though it will generally be true that there may be corresponding impulsive 

 screws and instantaneous screws, yet in the case where the principal planes 

 of the cylindroids are at right angles no such inference can be drawn. 



The case is a somewhat curious one. It will be seen that the calculation 



See Trans. Eoy. Irish Acad. Vol. xxx. p. 112 (1894). 



