Woodward — The Efficiency of Gearing under Friction. 101 

 If now we develop the first logarithm by McLaurin's 



fC7T 



theorem, we get a term which cancels out the term 



n x e 



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

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



. k TT 3 1+3& 2 7T* k(2+M 2 ) 7T 5 



•>„2 I" Q 



2n*e 2 3 n*e 6 



+ 



12 



n*e 



1 + 



15 



n*e 5 



+ 



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



-676 + & C - 



90 



n^e" 



The second fraction treated in the same way gives 

 ir 2 k'Tr 3 1 + Zk' 2 tt* k'(2 + 3k' 2 ) tt 5 



2n 1 2 e 2 3/i^e 3 



+ 



12 nfe 



4*4 



15 



7i 1 5 e 5 



+ &c 



7T" 



Adding these and withdrawing — j- 2 from the brackets, we 



get 



*-£+y? 



n^e" 



A;' — & 7r 7r 2 



""3" ^e + 6n x 2 e 2 + 



& 2 + /fc' 2 tt 2 2(k' — k) TT 3 



4 ?i , 2 e 2 



15 ?i 1 3 e 3 



&c. 



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



i? 



\Wj w,/ 



1 W 

 2 



271/ 

 3n n 



+ ^ 



7T< 



6n 2 e 2 



+ 



7T 



2/2 



[f 



2n 2 e 2 



2/2 



+ 



* 2 / 



2n 2 



4tt 3 / 

 15/i 3 e 2 



&c. 



(12). 



9. In formula (12) the values of n 1 and n 2 are to be found 



2r 

 as explained in § 7. The value of e = — <L is always unity 



r 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. 



