VOL. LXX.] PHILOSOPHICAL TRANSACTIONS. 683 



Carol, Putting q =: i, and n = p, as before, we have 



log- -~ = 2 log. -- — log. (,;^,)/_, . 



These theorems Mr. H. demonstrates, and then he adds some examples in 

 numbers, to show their utility in calculating the logarithms of several particular 

 numbers. 



XFIL On the Knowledge necessary for Judging of what New Kind of Sugar- 

 Cane Mill can be proposed. By M. Cazaud, F. R. S. From the French, p. 318. 



The first sugar mills, Mr. C. says, have been constructed without principles ; 

 and that few or no essential changes have been made. He says the common 

 horse mills have their diameter from 45 to 55 feet, suppose 30 at a medium. 

 To this mill is attached a couple of mules, and the bruising cylinder is about 18 

 inches in diameter ; so that the resistance of the canes is at Q inches distance 

 from the centre of action. It requires, to overcome this resistance, the con- 

 tinued effort of 4 mules, applied to a lever of about 24 feet ; which effort he 

 estimated at 600lb. or 150 lb. for each mule. In 24 feet there are 32 times Q 

 inches, or 32 times 600lb. ; so that the resistance of the canes is equal to about 

 1 9000 lb. in such a mill. 



If it were only required to overcome (no matter in what time) this resistance 

 of 19000 lb. we might conceive it to be by applying the continued effort of 2 

 men, estimated at about 50 lb. at the extremity of a lever of 388 feet, and then 

 we should have a product equal to that of the 4 nmles of the common mill ; 

 but then we should only obtain in 12 hours what we have in one only with the 

 4 mules. 



It must be known also that the common mill, for the use of which it is 

 necessary to devote at least 36 mules, gives only, one hour with another, but 

 above 80 or 100 gallons of liquor; and that a good water mill, such as is neces- 

 sary to make 250 or 300 barrels of sugar in the good seasons, ought to give 160 

 or 200 gallons per hour, on an average. It must also be known that, to give 160 

 gallons of liquor in the dry time of March or April, the cylinders which press 

 the canes ought to turn 2^ times in a minute, the same as to give 200 in the 

 month of January. 



It must be remarked also, that the difference in the product of the 2 mills 

 above-mentioned, sujjposing necessarily an equal difference either in the resist- 

 ances overcome, or in the times employed in overcoming them ; the resistance 

 which would be to overcome in a new mill, which should be equivalent to an 

 excellent water mill, ought to be supposed of about 38000 lb. or the moving 

 power to overcome it, ought to run through in 1 hour, the space which the 

 mules, in the common mill, run through in 2. 



4 s 2 



