of the Turning-lathe to a given Length of Band. 249 



or to V (4 — d») —^(1 — d) — WflT. 



Wherefore, the excess of the length of the band above the cir- 

 cumfereDce of the pulley, is 



V(4-<?) + d(i + .), 



and its excess above that of the fly-wheel, is 



V(4-d')_<f(|_«). 



Each of these expressions is dependent only on the difference 

 df between the diameters w and p. 



In the actual computation, we observe that 4 — (?= (2 -f- d) 

 (2 — d)\ and also that Jtf = secO.^. sin 5 when ^. sin d is 

 very small. One example of the entire calculation I subjoin. 



d=1.77; 



Log sec 6' 

 Log 124 



sin ^ = .885, 



15'= .33197 



= 2.09342 



2.42539 



2 + d 

 2 — df 



= 3.77 



4 — d' = .8671 



866761 = .931« 



3808 



3725 



83 



error 124. 



62° 



15' 



Log corr. 



= 1.0821041 

 = 43633 



= 266 



= 1.0864940 

 = 1.5707963 



C+0 = 



={ 



2.6572903 



1.7601032 



.1860103 



<-0 = 



{ 



0.4843023 



.3390116 



339012 



da->) 



Excess of band above pulley 

 Excess of band above wheel 



= 4.7034038 

 = 0.9311820 

 = 0.8572151 



= 5.6345858 

 = 0.0739669 



After this manner, I computed the excesses of the band, cor- 

 responding for a while to each tenth, then to each fifth, to each 



JANUARY MAUCH 1831. R 



