320 Proceedings of Royal Society of Edinburgh. [sess. 
drawn from that intersection, upon the revolving line traces the 
constrained locus. 
7. The extension to similarly constrained screws in a rigid 
system having otherwise three degrees of freedom is easy. The 
pitch is given by 
p (& 2 + w + &) = ipp 2 + wi y 2 + &f| 
or 
jpSp</>p = Spi/^p 
where p = xi + yj + zk , 
4>p = - + k£Sk)p , 
XP = - (ipftt +jp 2 Sj + kp 3 Sk)p , and 
= X^> * 
The cone Sp^ - 7r<£)p = 0 gives the directions of screws of given 
pitch. 
To determine the locus of screws of constant pitch, let a be a 
unit vector along a generating line of the ruled surface, and let cr 
he a vector to a point upon the line. We have 
crSp</>p = - Pi£V iaSia - ppf^j aS/a — p 3 £ V koSka + Vo. 
where v is a variable scalar. So 
o"Sp<£p — Vif/aa + Va , 
= V (if/ — p)aa + Va , 
a — “ (^-p) _1 Vo-aSp</>pl 
= - -p)a-y<ra = ^-^aS<r(^ - p)or . 
l/lp flip 
Thus 
$ 2 p<f>pS(r(iJ/ —p)cr = 
= (Pl£ -P)(P 2 V -P)(P£ ~P) • 
When = 0 this becomes 
S' 2 p</>pSo-i/ro- =PiP-iPffC , 
the surface upon which lie the rotors of the system. If we choose 
Sp^p = it becomes 
Scrif/a- —PiPiPz • 
( Issued separately October 7, 1902.) 
