64 KANSAS UNIVERSITY QUARTERLY. 
RELATIONS COMPLEX. 
The relations of the problem are so intricate, made so by the 
fact that one belt must fit all pairs of steps with the same degree of 
tension, that it is impossible to figure correct diameters for cones, 
arithmetically; or to calculate them at all, except by rules and for- 
mule which are merely approximately correct. 
FOLLOWING RULES, NON-APPROXIMATE. 
The rules here given are the only ones which permit an entirely 
complete treatment of all cases, with resort to approximations. 
They are based upon a non-approximate demonstration and will 
be found to give full and satisfactory results under all conditions. 
RULES TESTED. 
To thoroughly test the reliability of these rules in practice the 
diameters for the cones of the foot lathe, shown in Fig. 10, were 
taken from the diagram carefully and a circular disc of heavy bris- 
tol card board was cut out accurately to the one one-hundreth of 
an inch for each step of each cone of the pair. Then the two discs 
of a pair were fastened to a large drafting board by means of a fine 
needle through the center of each, the needle points being accu- 
rately set a distance apart equal the distance between the centers 
of the cones on the machine. Then a length of very fine copper 
wire was placed around the two, representing the belt, and was 
drawn up to just enough tension to be straight between pulleys. 
Then each one of the pairs of discs, representing the pairs of steps, 
was substituted for the first, in turn, with the distance between 
centers exactly the same and the same fine wire representing the 
belt tried on each. The tension of the wire was tried in each case 
by ‘‘snapping” it between the pulleys. The result was that the 
tension of the wire was so nearly the same on each pair of steps 
that no appreciable difference could be detected whatever. And 
the length of the belt measured from the wire was the length shown 
by the diagram to within two one-hundredths of an inch. 
For three years the rules here given have been used on designs 
of various characters with perfect satisfaction, both by the writer 
and others to whom they have been given. 
OPEN OR CROSSED BELTS. 
A design may call for a pair of cones with an ‘‘open” belt, which 
is usually the case; or a ‘‘crossed” or ‘‘twisted” belt may be re- 
quired, 
