50 KANSAS UNIVERSITY QUARTERLY, 
JG=JR—GR=LS—GR 
d 3) ase 
—=— — —(a sin a+ COS a). 
T T 
This is assuming | to be known, when, in reality, it is involved 
in the other quantities, but this may be done while establishing the 
relations. 
We now have the first two terms of the expression for R and rf, 
: ‘ ‘ dig fe s 
shown in the length JG. So if now we add >Z sme to JG, and 
then subtract it, we have then the lengths of two lines representing 
pies doe, 
R and r, respectively. This term 5 Sina may be had at once 
: : ae I i 
for any value of a by simply drawing a semicircle upon — AB as 
; 5 2 
diameter, as shown in Fig. 3. The intercept AT of the radius AP, 
Mitiad ts d= 
at any position 1s then — sin a. 
2 
[This step differs from the corresponding one in Reuleaux’s 
work, the result being the same. The semicircle is more satisfac- 
tory, giving at once —~— sin a without drawing any additional lines. ] 
2 
z Git 4 ‘ 
ay ott AT== 5 sin a upward from G, and downward from G, 
thus determining points U and V, when we have 
[pUeaIne 
and Neate 
1 being the correct half length of belt for these radii. 
Now similar points may be gotten in just the same way for other 
values of a. And when all values possible have been used the re- 
sult will be the smooth curve FUXVH, tangent to WE at X, which 
gives all possible pairs of radii which run together for any length 
of belt, the distance between centers of shafts being known or as- 
sumed equal d. 
THE COMPLETE SOLUTION. 
We now have in Fig. 3 a complete solution of the general prob- 
lem for the cases of open belts, perfectly correct and complete in 
every detail in so far as the analysis is concerned, though as may 
be seen not yet of a form suitable for actual use. 
Any length of belt as 1, in the figure, may be assumed, and the 
reference line JS quickly found, when the pair of co-ordinates to 
