52 KANSAS UNIVERSITY QUARTERLY. 
chosen to conform to the value of d on a permanent plotting, 
which would be inconvenient. 
II. We cannot choose the first pair of radi in predetermined 
ratio conveniently on the diagram. This must be done aside 
arithmetically, or by a separate figure. 
III. Having this first pair of radii, we can find the reference 
line and the length of belt, but only by a tentative operation, as 
has been explained. The difference of the radi (R—r), must be 
taken in the dividers and tried at various positions, till the location 
of a vertical is found such that this (R—r) laid off on it will just 
fit the two branches of the curve, as VU, Fig. 3. 
IV. This condition is fully met. We have all the possible 
pairs of radii for the length 1, found, easily attainable. 
V. But this, the most important of the conditions, cannot be 
met at all except by many trials each time, and herein is the 
chief objection to the use of this figure. It 1s very necessary to be 
able to find succeeding pairs which shall be in definite ratio. 
To overcome as many of these objections as possible, Fig. 3 1s 
transformed, in the Reuleaux discussion, by replotting in such a 
way as to make it possible to obtain any number of successive 
pairs of radu, which shall be in predetermined ratio when once 
the reference line is fixed. But this transformed diagram embodies 
the two serious objections which have already been made to it, 
besides sacrificing the length of belt 1, which, while not essential, 
it would be well to have retained: (1) It requires an irregular 
curve plotting, which necessitates either a permanent diagram, 
with the inconvenience of finding R and r in proportional parts, of 
d, or else a replotting each time:- (2) -The reference lime Won 
sucessive pairs of radii cannot be determined from the first or given 
pair, except tentatively, unless the case happens to be that of cones 
alike, with an odd number of steps, so the middle pair, with ratio 
1:1 may be that taken for finding the reference line. 
Considering closely these points of objection, and bearing in 
mind clearly just what is desired, it is seen that the one feature 
which is essentially unsatisfactory in both Fig. 3 and the 
Reuieaux transformation, is that both the radii of the pair R and 
r are measured in the same direction, and along the same line. 
This makes the finding of an R and an r of definite ratio impos- 
sible in Fig. 3, and not as convenient as might be in the trans- 
formed figure. 
If only R were measured horizontally, and r vertically, then it 
would be a simple matter to attain a desired pair of radii in definite 
ratio, This gives the final clue to the satisfactory form of diagram. 
