250 Mr Sang on the Tuniing-lathe, 



fourth, and to each alternate difference of diameter ; narrowing 

 the intervals as the rapid change of the differences required, 

 until the last forty numbers were computed strictly. The 

 somewhat irregular intervals that were thus left, I afterwards 

 filled up by the method of differences, taking care to leave no 

 chance of an error in the sixth, hardly indeed in the seventh 

 decimal place. The numbers which I have given are only to 

 five places, and may thus be relied on as true to the nearest 

 hundred thousandth part of the distance between the centres ; 

 except, perhaps, in one or two instances, when the rejected fi- 

 gures were 50, 49? or 51, and when it was difficult to say whe- 

 ther the last figure of the five should be preserved, or increased 

 by unit. 



I need hardly observe, that when d is supposed to be nega- 

 tive, 6 becomes so too, and that thus the two partial formulae 

 are but different cases of a single one. There are properly, then, 

 not two tables, but, if the numbers in the third column be con- 

 ceived to be written beginning at the last, and proceeding back- 

 wards to the top of the second column, and then returning down 

 the second, while the numbers in the first column are made to 

 run from — 2.00 to + 2.00, the whole of the results form one 

 series. 



If, in the lathe with crossed bands, w and p denote the dia- 

 meters, s their sum, we have sin fl = - ; the length of each free 



part of the band J (^ — — ) : the length in contact with the 



wheel w (^'\- 6\ and of that in contact with the pulley 



w; (^ 4- d) ; wherefore the whole length of band is 



which is exactly the formula, changing s into rf, for the num- 

 bers entered in the second column. 



If, then, we enter the first column with the sum of the dia- 

 meters of the wheel and pulley (plus twice the thickness of the 

 band), the opposite number in the second column will give the 

 length of the band when crossed. 



