52 Proceedings of the Royal Irish Academy. 



If the cylindroid reduce to a plane, then vi = 0. But if this is the 

 case, both of the terms in the expression of »»" must be zero ; for as ( V\i\tf 

 is negative, neither of the terms can ever be negative. Hence, we must 

 have 



SKiX^iX^S^tzX, + Ar«/«,Ai) - ArA,' (S^,X. + «SjujA.) = 0, (82) 



Aj=S^,X, - Ai'S/ujA: = 0. (83) 



The fii-st of these means that (/«iAi) and (/ijAj) must intersect, and the 

 second means that their pitches are equal. 



If (/iiAi) and (/ijAa) be a pair of screws which determine a canonical 

 cylindroid, then from the fonnula; already given it is easy to show that the 

 length of the axis of the cylindroid is 



^■i S^XS^,\^ ( FA.A,)' - (A.'^/i,A. - X,'SM.\,)' )i 



r (FA.A,)'(SA.A,)' )• ^^*^ 



It may Ije here remarked that in general any two cylindroids can be so 

 placed that all the screws on cither are reciprocal to all tlic screws on the 

 other. 



This condition will be secured if each of the two screws of zero-pitch on 

 one of the cylindroids intersects both of the two screws of zero-pitch on the 

 other, for as two intersecting zero-pitch screws are reciprocal, the condition 

 stated provides that each cylindroid shall contain two screws reciprocal to 

 the other. 



y. — Use of QttaUmions in the Theory of Reflected Screws. 



The subject of reflected screws has been already discussed in these 

 memoirs,* so I need here only repeat that if a standard plane be taken, 

 and if to the reflection of any screw from that plane a pitch be assigned equal 

 in magnitude but opposite in sign to the pitch of the original screw, then 

 the screw so formed is said to be the reflection of the original screw. 



The methods of quaternions present the vector coordinates of a pair 

 of reflected screws with extreme simplicity, as is shown by the following 

 statement : — 



It (fi, A) be the vector coordinates of a screw, then the reflection of that 

 screw from the plane Spi = has (/*', A') for its vector coordinates where 



IJL = - i/ii and A' = iXi. 



We first observe that if two lines intersect at right angles their reflections 

 will also intersect at right angles, for as the distance between two points is 



• Truu. Boy. Ir. Acad., vol. xxxii., pp. 119-127. 



