60 KANSAS UNIVERSITY QUARTERLY. 
involute, ENG. Drop the perpendicular GK and on AK as 
diameter describe the semicircle AFC. 
Now a, the angle of the belt, may be allowed to vary between 
its limits, o and go°, and for each value of a taken, the new origin 
A’ and the value of the constant (R-+-r) can be seen at once. 
And then the constant length of belt for this new origin, and value 
of (R-+r) can readily be had and plotted along the new horizontal 
reference line through A’ for the proper point on the belt line, as 
follows: 
Consider any value of a, as / PAE. PH is now the value of 
(R--r), which must be constant. For PH=d sin a, and we know 
d sin a=(R-+r) from Fig. 2. 
Projecting P across to J, drawing JQ, or 45°-line, and dropping 
QA’, a vertical, down to the 45°-line AK, we have A’ for the new 
origin and A’L as the new reference line upon which it will be 
convenient to find the value for 1, for this particular (R-+r). 
PN, perpendicular to AF at P, gives the point N on the involute, 
through which drop the perpendicular NM. Then, as in the 
former case, AM=d(asina+cosa) which is the first term of 
equation (g) and FK is the other term, being 
FK=d * sin a; 
2 
since AK=—BG=—d = ; 
and FK=AK sin a: 
Svaane FK=d - sin a, 
Adding FK to AM we have point S, which gives T, a point on the 
belt line, from which the half length 1] is shown as BV. 
Repeating this for succeeding values of a we have the curve 
WTD resulting, from which the length | can be obtained for any 
position of A’. 
This curve is found to be of very gradual] curvature, slightly 
convex downward. It is evidently limited by point W, vertically 
over E, and on a horizontal line through X, since the minimum 
value of 1 is d, when (R-+r) vanishes, bringing A’ to X. 
MODIFYING THE BELT LINE. 
Examining the deviation of this curve from a straight line join- 
ing P and D, it was found after a series of trials that by the follow- 
ing slight addition to the diagram the straight line WD may 
replace the curve: 
