54 Proceedings of the Royal Irish Academy. 



as this vector coincides with its reflection ipi = /), or 



.„ u, . SaX'^i .. . -^ iu,i SuX'H .. , 



'^ X SXi iM Ski 



thus verifying that the two screws intersect in the plane of reflection at the 

 point whose vector is p. 



We may note that the reflection of a canonical cylindroid is also a 

 canonical cylindroid. 



This is obvious from the fact that in the canonical cylindroid the bound- 

 ing screws are the screws of zero pitch, and that the reflection of a screw 

 of zero pitch is also a screw of zero pitch. 



But it may be useful to verify this from the expressions previously obtained 

 for the coordinates of a reflected screw. We have already found that, if 

 (Xi/ii) and (X,/ij) be the coordinates of two screws, the cylindroid they define 

 will be canonical if 



(SAiXj(<S)iiAj + SftiX,) - Xi'fS/ijAj - Aj'jSyiiAi = 0. 



As to S(fXi\i + /ijA,), we have already seen that the effect of substituting the 

 coordinates of the reflected screw is merely to change its sign. jS'AiAa is 

 unaltered, and - Ai'iS/zaA, and - Aj'<S)uiXi both change sign, so that the 

 equation is satisfied by the reflected cylindroid if satisfied by the original 

 cylindroid. The cylindroid determined by the two paire of screws (/iiAi) and 

 (/ijA,) will be altered by reflection from any plane into another cylindroid 

 which will be identical with the original cylindroid (though, of course, 

 difl'erently placed) if the pitch of every screw on the reflected cylindroid be 

 augmented by the common quantity 



(y\ \ ^7 {"^^i^j (S^tfti + SXjfi,) - Ai'iS'Aj/ij - As'(SAi/u,). 



For the two principal screws of a cylindroid are reflected into the two 

 principal screws of the reflected cylindroid. The original pitches were p^ + m 

 and Pa - m. The corresponding pitches in the reflected screws are - p„ - wi 

 and - fo + m. The addition of 2p^ to these will bring the reflected pitches to 

 the original pitches. Note, however, that the screw of maximum pitch 

 reflects into the screw of minimum pitch. With this is connected the fact 

 that the screw of maximum pitch is left-handed witli regard to every other 

 screw on the cylindroid. But reflection changes a left-handed pair into a 

 right-handed pair, so that the maximum pitch must reflect into the minimum. 

 If ft, A be the vector coordinates of a screw before its reflection from the 

 plane defined by the equation Si(p -a) = 0, then, after leflection, the 

 coordinates of the screw become 



- {(i,ti + 2Sai. VM), i\i]. 



