PALMER: THE’ DESIGNING OF CONE PULLEYS. 7, 
ANALYSIS ‘FOR CROSSED* BELTS. 
THE DISCUSSION. 
For this case we may now follow a discussion analogous to that 
presented for the case of Open Belts, and derive a simple diagram 
for use. 
This case is not considered graphically at all in Reuleaux’s 
treatment of the problem, it being considered of minor importance. 
But since an easily constructed and very useful diagram for the 
purpose can be had readily, it would seem well to make the treat- 
ment perfectly complete, and deal with the entire problem by one 
general method. 
For this case we have, from Fig. 2, page 44, the equation (3): 
T 
l=d cosa +( . ta ) (R-Lr) 
and also, from the figure, 
(R-++r)= d sina; 
Substituting this value for (Rr) we have: 
7 ; ed 
l—d cos 7 +d a ) sin a (8). 
: 2 : 
We see now, as before noticed, that 11s constant so long as 
is constant, and a will be constant when (R-+-r) is constant. 
When a is o, then 
= 
and (R-++r)=o, that is each pulley vanishes to a point. 
When — 50 ; 
2 
ld 
and (R-+-r)=d, which means that the pulleys just touch. 
And this shows, too, that the maximum l, is the same as for the 
case of open belts, zd. 
So now we may proceed to draw a very simple diagram exactly 
analogous to that used for the other case. 
Draw AB, AD and BC, as before, making AB—d. 
Now since the sum of the radii is to be constant, and since this 
sum equals d, for | maximum, it is seen at once that a 45°-line 
through B, BE, will in this case give all the possible pairs of radii 
for the problem, in just the same manner as the circle arc BFE of 
Fig. 4, in the preceding case, 
