142 Mr Sang on the Friction of Time-Keepers 



Or, 



91 . . / . A 



sm 



A (sin-) 



(2.4)2.3.5 v 2 



23031 . / . A \ 6 



— sm A (sin- ) 



(2.4.6) 2 .3.5.7 V 2 / 



12 949 497 . k / . A , f 

 A (sin-) 



(2.4.6.8) 8 .3.5.7.9 ^ 2 



13 793 405 G25 A / . A v ro , 



^ (2.4.6.8.10)^.3.5.7.9. 0 SmA ( S11 ^) + ^ 



Here the order of the denominators is sufficiently obvious 

 That of the numerators can be exhibited thus: Let A, B, C, 

 D, E, etc. represent the successive numerators, then 

 A = 1 



B = A. 4 3 + 3 3 



C = B. 6 3 + (3.5)? 



D = C. 8 5 + (3.5.7) 3 



E = D. 10 3 + (3.5.7.9) 3 , etc . 



By means of which formulae, logarithms of the coefficients 

 have been computed. 



At the first glance, however, it will be apparent that the va- 

 lue of if/A given in terms of the radius will be exceedingly in- 

 convenient for computation, since it would be necessary to con- 

 vert the arcs also into decimal parts of the same unit. It would 

 be more convenient to take some small arc, as one minute, for the 

 unit ; in this case, all the coefficients would need to be multi- 

 plied by the constant ^~ , or their logarithms to be increased 

 by the logarithm of that fraction. The formula then becomes 



10800 I . / . A \ 2 

 ^A=A'-f — — ^sinA (sin- ) + etc ; 



and its application is given by the equation 



A'— A' 



T *J sec <f- 



•4/ A — ^'A 



n 



the accents denoting that the quantities are taken in minutes of 

 a degree. 



The computation of a table of the values of 4/ A being very 



laborious, the operations have not yet been revised. In this 



case it is deemed advisable to postpone their publication until 



the revision is completed. 



60. North Bridge, | 

 22rf Jtfay 1835. J 



