PALMER: THE DESIGNING OF CONE PULLEYS. BS 
This discovery that this curve, giving all possible pairs of radi 
for the problem, can be so easily drawn by a simple circle arc, was 
of the utmost value, making it possible to attain completely the 
desired form of solution at first determined upon. Now this can 
be drawn for any distance between centers, d, to any suitable scale. 
And it remains a comparatively simple matter to bring in the 
relation of the length of belt and to locate the reference line for 
any case, and then to fulfil each of the five predetermined condi- 
tions, page 42. 
APPLICATION OF NEW DIACRAM. 
If now we have a certain half length of a belt, 1, it may be 
applied just as in Fig. 3, by laying off along BC, and projecting 
down to the z-line, drawn as in Fig. 3. Then A’ is the new origin, 
for R and r, and all the possible values of R and r for this 1 are 
given by the curve between N and M; this may be easily seen from 
comparison of the two figures. In Fig. 3 the application of | takes 
off a certain length OJ from the value of both R andr, for maxi- 
mum |, leaving the correct values of R and r for this particular |. 
In the same way in Fig. 4 the same amount is taken from the 
values of R and the values of r, just as in Fig. 3, and the same 
amount as is there removed. And NM is the available part of the 
curve just as the portion in Fig. 3, contained between verticals, 
through Z and X, was there. 
Now it remains to reverse this and from a pair of radii find the 
length of belt, and then to follow out the steps set forth in the five 
conditions, page 42. 
If we have given a pair of radii, the new origin, A’, and hence 
the half length, 1, can at once be determined for it by the following 
simple steps: 
Lay off the R from A to H, and the r from H to K, and draw 
AK. Through K draw a 45°-line KS, finding S. Then through 
S draw back SA’, parallel to KA, thus locating A’, the new origin, 
and A’N and A’M the new axes. Producing A’M to the z-line, 1 
is seen at once. 
The reasons for this are plainly evident. The new origin A’ 
must be so located that this particular R and r will just fit the 
curve, when measured from it, that is be co-ordinates of a point 
S, of the curve, when A’ is the origin. We have simply located 
A’ so this will be true, having made triangle A’SO—AKH. 
Now with the new origin A’ determined, any other pair of radii 
suitable for this half length of belt 1, can immediately be had from 
