58 KANSAS UNIVERSITY QUARTERLY. 
The new origin for any particular value of (R-+r) may at once 
Cc 
be found and any other pair of radii having any desired ratio, 
quickly obtained by just the same steps as in Fig. 4. 
INTRODUCING LENCTH OF BELT. 
This does not, as yet, take into account the length of belt. The 
length 1, is virtually #wed when the new origin A’ is located, but 
its value is not shown. As | is practically of minor importance 
the figure, thus far, would be sufficient. It is, however, convenient 
to have the belt length shown at once, and it should be shown for 
the sake of an entirely complete treatment. It is interesting, also, 
to note the form of the belt line in this case. 
It will develop that here, just contrary to the other case, the 
line giving the radii, is a straight line while the ée/¢ line is a curve 
instead of the straight ‘‘w-line” of Fig. 4. 
This curved belt line may be plotted for the purpose of studying 
its characteristics, in order to find if possible some simple equiva- 
lent for it. 
To plot this belt curve we may proceed as follows: 
Consider the expression for |], re-writing equation (8) we have: 
]==d(a sin a+ cos a)+d = sin a (9), 
the terms of which may be obtained graphically in a way exactly 
similar to that employed in treating the case of open belts. 
In Fig. 6, strike the arc BPE about the center, and draw the 
