194 Description of a Machine for raising Coals 

 denjcik, and frequently causing the immediate destruction 

 of the men who iriay be ascending the pit at the time, or 

 dashing to pieces the basket and its contents. 



Besides the unnecessary expence arising from the use of 

 hempen ropes, and the breakage of chains when apphed in 

 the common way, the forms of the barrels are quite erro- 

 neous in principle. Some are cylindrical; others formed of 

 frustums of cones united at their bases, without any deter- 

 minate proportion in their perimeters, or regard to the weight 

 of the rope or chain working thereon, both of which are 

 absolutely necessary to acquire a maximum effect. 



The convex surface of a frustum of a cone, is = to the 

 convex surface of a cylinder of the same altitude, having its 

 circumference = to half the sum of the perimeters of the 

 frustum : and circumferences of circles being to one another 

 as their diameters, the surface of a barrel formed of two 

 frustums of right cones (united at their bases), each 64 

 inches diameter at one end, 32 at the other, and 34 long, 

 which is the size we have adopted here, is = to the surface 

 of a plain cylindrical one, 48 inches diameter, and 108 long. 

 Each will therefore bend the same length of cordage in an 

 equal number of revolutions, and so far they are equal to 

 each other; but they vary very considerably in the momenta 

 required to work them. 



Let a = the weight of the basket of coal, and b = that 

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

 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 + b= the counterpoise required 

 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 — b = the counterpoise required at the same 

 radius. 



On the barrel formed of frustums of right cones, when 

 the load is eased from the bottom of the pit, it and the chain 

 are suspended from one of the smaller perimeters (the op- 

 posite chain being bent on the barrel), -^ + ^ = the coun- 

 terpoise required at 32 inches radius; and when it is wound 



to 



