Ball — Contributions to the Theori/ of Screws. 667 



by a screw in correspondence with the variation of one of its co-ordi- 

 nates. 



Let the six screws of reference be 1, 2, 3, 4, 5, 6. Form the cy- 

 lindroid (A, 1), and find that one screw -q on this cylindroid which has 

 with 2, 3, 4, 5, 6, a common reciprocal. Let the adjoining figure be 

 a pencil of four rays parallel to four screws on the cyKndroid. Let OA 

 be parallel to one of the principal screws ; OX be parallel to A, Orj to -q, 

 and Ol to the first screw of reference. Let the angle A Ol be denoted 

 by A, the angle A Or] by B, and the angle A OX. by 0. To find the 

 component Aj we must decompose A', a twist on A, into two component?, 

 one on t], the other on I. The component on r] can be completely 

 resolved along the other five screws of reference, since the six form one 

 system with a common reciprocal. If we denote by ?/ the component 

 on 7], we then have 



sin(^ - A) sin(</) - £) sin(<^ - A) ' 



and if a and i be the pitches of the two principal screws on the cylin- 

 droid, we have for the pitch of A the equation 



p = a cos^ 6 + b sin^ ; 



cause tl 



the screw along this cylindroid. 



AT X ,sin(0-5) 



Now Ai = 7]' ^-J -(, 



sin(0 -A) 



and as the other co-ordinates are to be left unchanged, it is necessary 

 that 7]' be constant, so that 



^_ ,sin(^-^) 



^""^ sin-(<^-^)' 



and hence -f- ^(b-a)sm Id, ■ , . ^ -- -'-. 



dXi 7] sm.{J5-A) 



Also — - = — .^=7- =- cos(^ - ^). 



dXi d4> «Ai "• ^ 



Hence, substituting in the equation 



^,dp ^ dX' 



we deduce a = h tan ^ tan A ; 



but this is the condition that A and the first screw of reference shall be 

 reciprocal {Screws, p. 37). 



also -^^ = -J- . -^, because the effect of a change in Aj is to move 

 dXi d(p dXi 



