PALMER: THE DESIGNING OF CONE PULLEYS. 45 
x I—di eosia 
= — 5 
DY (R-+-r 
SY Saat ele 1—d cosa 
Cc == = ———— J 5 
lo tae) 
|= dcosja\ a ket 
Rr i= aie 
and 
sin =cos{ 
Set i174 d2=2ER4 ue) 
———-——COS]| __ ee . ( ) 
d ( Ree ‘ 
An inspection of equations (3) and (4), and Fig. 2, shows that 
this case of crossed belts does not present the difficulties of the 
other, and far more important case of open belts. For, if (Rr) 
be kept constant, a, and hence 1, will be constant. This means 
that we may proportion the first pair of steps of the cones by the 
simple arithmetical rule, and then any other pair of radii whose 
sum is equal to that of the first pair will serve for the radi of an- 
ether pair of steps which the same belt will fit. 
The length of belt is not thus given, but as this is of minor im- 
portance the proceeding just outlined would answer sufficiently well 
for all practical purposes. If neither case presented greater diff- 
culties, there would be no occasion for an extended treatment of 
the problem. But as the case of open belts necessitates a special 
graphical process, the simpler case of crossed beits will be included 
in the same method, and a useful diagram analogous to that required 
for the first case presented. 
COMPARISON OF EXISTING METHODS. 
As the general relation of equation (2) for the case of open belts 
offered no possibility of a direct solution of the character required, 
a comparative investigation was made of the various ways in which 
the problem has been attacked, and of the character of the methods 
proposed by leading authorities. 
The following is a reference list of some of the best methods 
heretofore used for treating the problem, with a brief note as to the 
character of each: 
Unwin’s ‘‘Elements of Machine Design” (new edition), page 
373, calls for tedious calculations with approximate formule. 
Rose’s ‘‘Modern Machine Shop Practice,” gives tables for find- 
ing the radu of the steps. Unsatisfactory to use. 
