138 Mr Sang on the Manner in which 



The integration now needed for the determination of t, can 

 be performed in a variety of ways, fitted for different species of 

 inquiries : the usual substitution 



A — (p 



: Sill ■ 



best suits the pi-esent. Computing from this the value of ^a in 

 terms of^e, substituting and simpHfying we obtain 



/• I i / . A — ?>\2 . '-^ 1 ~% 

 lt=.k i 1 — I sm — - — I sine V « e, 



V g&ccip \ \ 2 / J 



which expanded by the binomial theorem, becomes 



/ I ( 1 / A — ip\i . 1 1.3 / . A — ip\« .4 i 



lt=\J < 1 +7r( sm — -— ) sine +-— tsin I sine +etc >J<f . 



^ gsec(p {_ 2\ 2 / 2.4 \ 2 / J 



Each term of this series can be integrated by means of the 

 formula 



/' • ry^" . ^ n—ip. „_2 . „ 1 ,. . ^.n—l 



smZ a Z = / sin dZ — — cos Z smZ 

 n J n 



After the proper arrangements the value becomes 



--!(n)X-^)*-(SI)X-^-^A-} 



COS e sin e" | (^;^)'(sm ^)V etc ' 



_ cose sine 



2 



■ — cos e sin< 



3.5 [ \2.4.6/ \ 2 y ■ '" ) 



+ etc. 



This formula enables us to compute the time of the arrival of 

 the pendulum at any point in its path, the time being reckoned 

 from the instant of maximum velocity. It would be easy, were 

 it worth while, to shew how it could be applied to the computa- 

 tion of the influence of a given escapement on the rate of a clock. 

 The preceding is analogous to that already given for watches. 

 For the purpose of aiding these computations, the logarithms 

 of the coefficients arc given in Tables I. and II. 



