Woodward — The Efficiency of Gearing under Friction. 101 



If now we develop the first logarithm by McLaurin's 

 theorem, we get a term which cancels out the term — 



and the remainder is exactly divisible by 1 + k 2 . The 

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



7T 2 h _tt*_ 1 + 3A; 2 7T* k(2 + Sk 2 ) tt 5 

 2n^e 2 + 3 n * e * + 12 ' ^e 4 + 15 nfe 5 + 



2 + 15k 2 + 15& 4 7T 6 

 90 <? + &c ' 



The second fraction treated in the same way gives 



7T 2 k'7T* 1 + 3k' 2 7T 4 fr(2 + 3fc' 2 ) 7T 5 



2n 1 2 e 2 Sn/e 3 ' 12 n x 4 e 4 15 n/e 8 



7T 2 " 



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



7b. 6 



get 



B-i 1 ± i vA* k ~ h * 



1 lto 2 e 2 ' + 



tf + lc' 2 it 2 2(JC' — JC) 7T 3 



4 n x 2 e 2 15 n*e 3 



but k' — k = 2ef t k 2 + k 2 = 2 (1 + e 2 )/ 2 , hence 



-£+*)¥[' 



3/1, ' 6^ ' 2n 1 2 e 2 





+S--T& 



9. In formula (12) the values of n x and n 2 are to be found 



as explained in § 7. The value of e = — °- j s always unity 



*i 

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

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

 •' &c. ,, covers only very small quantities. The common 

 approximate formula stops with the first term of the series. 



