138 Mr Sang on the Maimer 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 — rip . A — <p 

 sin — - — = sin — - — . sin e 



best suits the present. Computing from this the value of in 

 terms of ae 9 substituting and simplifying we obtain 



/• 1 f / . A — <p\ 2 . 2 1 ~\ 



o i— U -< 1 — sin — - — 1 sin*? \ 



V ^secp (_ V 2 / J 



which expanded by the binomial theorem, becomes 



u =^ { 1 + ¥ ( sin ^ir ) 2 sin e?+ 2 L r( sin Y sin /+etc } u - 



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

 formula 



/sin Z n HZ = 5 -^— - f sin* -2 2 Z — — cos Z sin 7?~ X . 

 After the proper arrangements the value becomes 



.cose sine (^"^C 3111 ~^ =: ^) 2 ~ h C^tI^C 8111 "i^) 4 (^tI^XC^ 11 "T"^) 6 " 1 " etc }* 



-|cos,sin^{ (^)t sin ^) 4 + (^)X sin ^? i ) 6+etc } 

 2.4 • 6 ( /1.3.5\ Y . A — p\ 6 i 



~ sTo cos * sm * \ Kure) ( sm V) * etc j 



4- 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 are given in Tables I. and II. 



