17 
A 
P CL\ iC I 
LU IfJ -.v-v^V 
h ' 
I N \; 
\ 
X 
a I W 'a 
I I S', 
Thus the vertical cross 
planes through X fix the 
circles for winding off from 
aici at X and on to (3i b at 
Y, and the vertical cross 
planes through Y fix the 
circles for winding off from 
the j3i b at Y and on to a x a 
at X h Still the windings off* 
at X and on at Yi cannot 
be windings of the same 
stretched string as the wind- 
ings off at Y and on at X h unless X lie in the same circle 
as X\ and F in the same circle as Fi; since by the bare 
turning of a drum a stretched string can only be guided 
from point to point in the same circle. That the same 
stretched string therefore may so engirdle the drums as 
always to wind itself off at the same fixed points X and F 
and always to wind itself on at the same fixed points X x and 
Y 1 it is needful and it is enough that the vertical cross 
planes through X be the same as the vertical cross planes 
through F. And this again happens just when the pairs 
of cross planes have the very same common section, or 
again just when the points X and F lie in the same 
vertical straight line. 
Hence through the lines &Cc and §iA§ draw vertical 
cylindric surfaces. Let P be the common section of these 
surfaces. Then planes drawn through the vertical straight 
line P perpendicular to the axes AiOA. and BiOB are the 
planes of the pulleys. 
If the axis be at right angles to one another (or rather, 
if from the forelook OA of the lower axis the forelook OB 
of the higher axis have a bearing with a right-hand- ward 
start either of once or of thrice a right angle), the lines 
C\Cc and are severally the straight lines aid and j3i/3< 
