Ball — Kinematics and Di/namics of a Rigid System. 257 



If and q be the pitches of two reciprocal screws, then they will 

 continue to be reciprocal if the pitches be changed into 



and ^ • 



1 + pm 1 - qm 



respectively, where m is any quantity whatever. This is obvious from 

 the consideration that the function 



I + pq 



remains unaltered when p and q are replaced by their altered values. 

 If therefore we have, two reciprocal systems of screws, and all the 



pitches of one system p be changed each into , , where m is the 



1 + 2)ni 



same quantity for all the screws, then each screw will continue to be 



reciprocal to all the screws of the reciprocal screw-complex where each 



pitch a is replaced by . It hence follows that all the screws of 



^ I - qm 



an w-system will continue to form an ?i-system if the pitches be all 



changed by writing, instead of each p, the more general value ~ . 



1 + ptn 



We may express this in a very interesting manner by the statement 

 that — " If all the pitches of a screw system be homographically trans- 

 formed, subject to the condition that the pitches of + 1 and - 1 remain 

 naltered, then the modified pitches are also a possible pitch distribu- 

 tion." 



We may note that this change cannot affect two polars whose 

 pitches are reciprocal, for, if p and q be reciprocal magnitudes, then 

 so are obviously 



p + m ^ q + m 



and . 



1 + mp 1 + qm 



We are now enabled to take a remarkable step in the simplification 

 of the question of pitch distribution on the surface which replaces the 

 cylindroid in elliptic space. We have, as before, 



p = tan (a + (f>) tan {(3 + 0) = tan (a - <^) tan (/? - 0), 



or, by transformation, 



tan a tan (3 + tan tan cf) 



P 



1 + tan a tan (3 tan 6 tan <fi' 



