— In  many  cases  which  occur  in  practice  the  axes  of  the  two
screws  are  perpendicular  to  each  other ;  so  that,  in  ^g,  110,  page  1 60,
AI  P  and  alp  are  at  right  angles,  I  0  coincides  with  Ip,  and
I  c  coincides  with  IP;  and  therefore  the  divided  axial  pitch  of
either  screw  is  equal  to  the  drciUo/r  pitch  of  the  other.  In  such  cases,
and  especially  where  the  diameters  of  the  pitch-cylinders  are  very
unequal^  so  that  the  larger  screw  is  called  a  toorm-wheelj  and  the
smaller  an  endless  screw,  it  is  often  convenient  to  design  the  traces
of  the  threads  on  a  plane  normal  to  the  axis  of  the  woim-wheel,
and  traversing  the  axis  of  the  endless  screw;  and  then  it  is  evident
(as  Mr. Willis  appears  to  have  been  the  first  to  show)  that  if  the
traces  of  the  tlumds  of  the  worm-wheel  be  made  like  those  of  a
spur-wheel  of  the  same  radius  and  pitch,  and  those  of  the  threads
of  the  screw  like  the  traces  of  the  teeth  of  a  rack  suited  J^^^j?;>r


164


GEOMETRY  OP  MACHINERT.
"with  that  spur-wheel,  the  worm- wheel  and  screw  will  gear  correctly
together.
Fig.  112  represents  a  worm-wheel  and  endless  screw.
The  lower  part  of  the  figure  is  a  diagram  drawn  on  the  common
tangent  plane  of  the  pitch-cylinders.     I  is  the  pitch-iK)int;  I  C  is


KT


Fig.  112.
