42 KANSAS UNIVERSITY QUARTERLY. 
so much to be desired as just the right kind of a graphical con- 
struction. The form of this construction should be so simple that 
it can be drawn at once from memory for any particular case in 
hand, and with the simple drafting instruments, without the use of 
the irregular curve or any special appliance. And this construction 
must be so complete as to give a// of the desirable features of a 
solution for every possible case of the problem, directly and with 
exactness. And it should be based on a non-approximate analysis. 
These requirements are rigid ones, and ones which the general 
equation for the problem would seem to offer little promise of ever 
fulfilling. But it has seemed so highly desirable that there should 
be a solution conforming exactly to each of the restrictions named, 
that the question has been studied from every point of view in the 
determination to find a treatment which would not be a compro- 
mise in any particular, if such a treatment were in any way possi- 
ble. 
As the result, the present discussion of the problem is offered, 
with the deduction of a practical method, which, it is believed, 
embodies all of the désirable features at first determined upon. 
THE PROBLEM. 
The following is the problem: We should be able, 
I. To assume any distance between centers of the shafts; 
II. To choose the radius of a step on one shaft, and to get the 
radius of the corresponding step on the other, the two radii to be 
in a predetermined ratio. This can be done arithmetically, but 
should be included in the graphic process. 
III. From this pair we should be able to have, at once, the 
length of belt required for the two cones. This length of belt is 
now a constant quantity, and must fit all the other pairs of steps 
about to be determined. 
IV. Dependent upon this length of belt, we should now be able 
to obtain readily an indefinite number of pairs of radii, which will 
‘run’ to this determined length of belt with the same degree of 
tension. 
VY. From all these possible pairs which run to this length of 
belt, we should be able to select a certain pair of radii, which shall 
bear a definite ratio to each other. 
In addition, we have from practical considerations that the series 
of speeds should form a geometrical progression, except in cases 
where particular reasons exist for having a certain definite speed 
at each step. This geometrical series of speeds is readily obtained 
if condition V is fulfilled, 
