Woodward — The Efficiency of Gearing under Friction. 101 



If now we develop the first logarithm by McLaurin's 

 theorem, we <]ret a term which cancels out the term — 



and the remainder is exactly divisible by 1 + ^'^. The 

 result, which is the value of the first fraction, is 



7r2 Jc 7r=^ 1 + 3^-2 TT* A;(2+3^2) tt^ 



+ 



2n^^e^ 3 n^^e^ 



12 



+ 



+ 



n^*e* ■ 15 1 ^ 



2+ 15A;2 + 15^-^ TT^ 

 90 np 



The second fraction treated in the same way gives 

 7r2 k'lr^ l + SJc'^ TT* ^•'(2 + 3^''2) ir'- 



+ (fee. 



2n2e2 



3nj^e^ 



+ 



12 



n*e^ 



15 



?i,^e^ 



+ &c. 



Adding these and withdrawing —y~2 from the brackets, we 



Sfet 



^={l^+n)T 



h' — h IT tt' 



+ 



3 n^ Qn.^(P- 



4 n^^e^ 15 n^^e" 



but Jc' — k = 2ef, k^ + l\^ = 2(1 + e2)/2, hence 



— &c. 



i? 



VtIj 71,7 



2 



27r/' tt' 



1-^ + 



3n, 



+ 



6?t, e 



2^2 



+ 



.2^2 



2n 2e2 



ys 



47ry 



2rj2 



15??,'^e2 



&c. 



(12). 



9. In formula (12) the values of n^ and n^ are to be found 



2r 

 as explained in § 7. The value of e = - — 5- is always unity 



or less. The terms in the series are arranged in order of 

 magnitude for common values of n^, e, and/". The character 

 '* &c." covers only very small quantities. The common 

 approximate formula stops with the first term of the series. 



