PALMER: THE: DESIGNING OF CONE PULLEYS. 47 
THE REULEAUX ANALYSIS. 
THE DISCUSSION. 
The following is the discussion found in the translation of Reu- 
leaux’s ‘‘Constructor,’’ page 189, essentially as given there, merely 
amplified somewhat for the sake of clearness. It is remarkable in 
that instead of attempting to eliminate the ‘‘angle of the belt’ in 
some way, as is usually done, this angle is carried on through the 
discussion and used to a point where it easily disappears. The 
entire Reuleaux solution shows a wonderful insight into the rela- 
tions of the problem and extreme ingenuity in dealing with them. 
CASE I—OPEN BELTS. 
Referring again to Fig. 1 we have for this case equation (1): 
l=d cos ook +r)+a(R =i) 
and the equation, d sin a=R—r. 
Combining these we have the two equations for R and r, respec- 
tively: 
] ‘ lek 
R—— ———(a sin a+. cos a)+ — sina (5) 
7 Li ee 2 : 
aii... Gina 
————— (a Sina-+ COS a)—— SIN a, (6) 
7 T 2 : 
which differ from each other only in the sign of the last term. 
In Fig. 3 draw AD and BC parallel and at a distance AB apart, 
equal d of the above equations. Draw AB, leaving the length of 
the rectangle undetermined as yet. Then draw the quadrant BE, 
with radius AB—d. Now, within the limits of this arc BE will lie 
all values of angle a, of equations (5) and (6), which can occur. 
For, from a physical consideration of the matter with the aid of 
Fig. 1, it is readily seen that a is limited by 0° and go°. That is, 
at the limiting case in one direction, when the two pulleys are 
equal, or diminish to mere points, a—-o: and when one decreases to 
r—o while the other increases to R—d, then u—go*. Now let a 
have any value, as / EAP, Fig. 3. Draw PN perpendicular to AP 
Ate. €. tangent tothe arc BE at P) And make PN=-arc PE. 
That is for any value of a, N isto lie on the involute EF, of the 
arc BE. Drop the perpendicular PM to AD, and draw NK per- 
pendicular to PM. Draw RO through N, perpendicular to AD. 
Then from the geometry of the figure we have 
AQ = AM+MQ = d(a sin a+ cos a), (7) 
