18 
The cylindric surfaces are then nothing hut the vertical 
planes through a x a and (3i/3 and therefore cut one another 
at right angles in the straight line P. Then too the vertical 
plane through cwa not only touches the lower drum face hut, 
passing through P perpendicularly to the higher axis, is 
itself the plane of the higher pulley; likewise the vertical 
plane through |3ij3 not only touches the higher drum face 
hut, passing through P perpendicularly to the lower axis, 
is the plane of the lower pulley. So then the vertical 
straight line P touches each of the faces of the pulleys and 
each of the midcircles. And so the millwright rule here 
holds good — “ Plumh the centres of the leading off faces.” 
These cases of rightangled axes are alone dealt with in 
art. 185 of Willis’s Mechanism, and seemingly as if there 
were no others. 
In all other cases the vertical cylindric surface through 
c x Gc touches the vertical plane through aiCa along the vertical 
straight line G and in an unbroken sheet runs away endlessly 
therefrom on both sides between the parallel vertical planes 
through OiCa and AiOA toward the vertical plane through 
AiOA as an asymptote. Also the vertical cylindric surface 
through § 1 A S touches the vertical plane through (3i A j3 along 
the vertical straight line A and in an unbroken sheet runs 
away endlessly therefrom on both sides between the parallel 
vertical planes through ]3i Aj3 and BiOB toward the vertical 
plane through BiOB as an asymptote. The plate bounded by 
the parallel vertical planes through aiCa and AiOA is thus 
split by the vertical cylindric surface through CiCc into two 
slices having for a common bounding face the vertical 
cylindric surface through CiCc and for their other bounding 
faces severally the vertical planes through a±Ca and AiOA. 
And the plate bounded by the parallel vertical planes 
through |3iAj3 and PiOP is split by the vertical cylindric 
surface through Si A S into two slices having for a common 
boundary the vertical cylindric surface through Si AS and 
