360 Mr. J. Preecott on the 



terms on the right o£ (66) are small quantities, and all that we 

 had to show to justify (64) was that the whole quantity on 

 the right of (66) is small compared with the large quantity 

 o\ Thus (64) is justified. 

 56. From equation (64) 



approximately, since the fraction in the brackets is a small 

 one. We might carry the expansion further, but nothing 

 would be gained by doing so. We only want to discover 

 the type of motion, not the precise amount of it. Integrating 



(67) we get 



(7 /~o rf i(f~l)m -| 



»=±}W m2 -i-*y m >T-r f } dT 



= ± | v / m 3 T 8 -r/ + V^sin - 1 ^ 



- ii(f- 1) log (mT + Vm'T-rf) T + F, (68) 

 F being a constant. Also 



to the same degree of approximation as we used in (67). 

 Therefore, 



x{ 1+ 4iS^} M±i(/ - 1 ^ ±6 V ™ 

 where 



M = wT+\/m s T 8 — r/, -j 



5 = i/^sin- 1 ^ + s/m^-rf ) * 



The double sign in (70) gives two separate results, and 

 since the equation for % is linear the sum of the two results, 

 with independent constants of integration, is a value for ^. 



