256 Proceedlnga of the Royal Ii-ish Academy. 



The simplest way of defining the surface is as the locus of the in- 

 tersection of the planes 



{ Xi - X:i tan ^ = 0, 

 X2, — Xi tan <^ = 0, 

 with the relation sin 2a sin 26 + sin 2/3 sin 2^ = 0, where 



tan p = tan (a + cf)) tan {(3 + 6) = tan (a - ^) tan {(3-6) = the pitch. 



It is hence easily verified that a ray intersecting two conjugate 

 polars intersects also two screws of equal pitch. This is the generali- 

 zation for elliptic space of the well-known property of the cylindroid, 

 that a ray cutting a generator at right angles passes also through the 

 equal pitch screws on the surface. This theorem is thus seen to be 

 the survival of the more elegant and symmetrical theorem in elliptic 

 space. In the elliptic space no less than in the ordinary space this 

 theorem has important dynamical applications. 



For some time I found a difficulty in tracing in elliptic space the 

 analogue of the well-known property in ordinary space, that the pitches 

 of all the screws on a cylindroid could be augmented by a constant 

 quantity. At length however I noticed that this too was only a sur- 

 vival in ordinary space of a more complete theory in elliptic space. 



If two screws be reciprocal, and if p, and p' be their pitches, and 



X and y the lengths of their two common perpendiculars, then we 



have 



p + p' 



tan X tan y = -. 



■^ 1 +pp' 



If therefore two screws be reciprocal they will continue reciprocal 

 notwithstanding changes in their pitches, provided only that 



p+p' 



\ + pp' 



remains unaltered. 



It is hence easy to show that if all the screws of a cylindroid have 

 their pitches p changed into 



p + m 



1 + pm' 



where m is a constant, then they will still be admissible as a possible 

 pitch distribution. This somewhat remarkable result can be demon- 

 strated with equal facility for a system of the n"' order as well as for a 

 cylindroid, and we therefore give the more general demonstration. 



