MACHINE FOR RAISING COALS OR ORE. 113 



and 54 long, which is the size we have adopted here, is 

 equal to the surface of a plain cylindrical one, 48 inches 

 diameter, and 108 long. Each will therefore bend the same 

 length of cordage in a equal number of revolutions, and so 

 far they are equal to each other; but they vary very con- 

 siderably in the momenta required to work them. 



Let a = the weight of the basket of coal, and b = that of F«rce requited 



. with a cylux- 



the descending part of the chain; then, on the cylindrical der: 



barrel, when the former is hooked to the end of the latter, 

 and eased from the bottom of the pit (the opposite chain 

 being bent on the barrel), a + &— the counterpoise re- 

 quired at 24 inches radius; and when it is wound up to the 

 top (the descending part of the opposite chain hanging 

 down the pit), a— 6— the counterpoise required at the 

 same radius. 



On the barrel formed of frustums of right cones, when and with two 

 the load is eased from the bottom of the pit, it and the ^^"^^3 ™^ °^ 

 chain are suspended from one of the smaller perimeters 



(the opposite chain being bent on the barrel), \- — — 



the counterpoise required at 32 inches radius; and when It 

 is wound to the top of the pit, it is suspended from the 

 larger perimeter of one frustum, while the descending part 

 of the opposite chain is hanging down the pit from the smaller 



perimeter of the other, and in that position a — — zz the 

 counterpoise required at the same radius. 



Consequently, by supposing a, the weight of the basket This applied 

 of coal, to be SOOlbs. and b, the weight of the descending "^ P'^'^^^'^®' 

 part of the chain, 400lbs. (these are the weights which we 

 have adopted here), we have the counterpoise required upon 

 the cylindrical barrel, at 24 inches radius, 1200153. when 

 the basket of coals is at the bottom of the pit, and 400lbs. 

 when it is at the top; but upon the barrel formed of frus- 

 tums of right cones, the counterpoise required at 32 inches 

 radius is 600lbs. in each position. And as the counterpoise 

 required is in inverse proportion to the length of the radius 

 at which it is applied, we have 24 : 32 : : 600 : BOOlbs. the 

 countei-poise required upon the barrel formed of frustums 



Vol. XXI— Oct. 1808. I of 



