524 Dr. J. W. Nicholson on the Bending of 



Moreover, 



^(z) = i^ + ^cos«- j -a.sinaj-^^- ^ + - 4 -... j 



-0tana+£ 5 (tan«-itan 3 a), . . (22) 



where there exists the identity 



l+^+M 2 2 H- ... = (1 + X 3 #+X 6 # 2 + ..-)- 1 . (23) 



The next approximation to that of Lorenz becomes, on 

 selection of the proper terms, 



B,,=.r/(* 2 -m 2 )i 



<j) n z=±7r -f .zfcos ol— - —a. sina J + ^f sec a-f ^tan 2 asec«J. 



• • • (24) 



Lorenz has also shown that if m is greater than z, and 

 ?n— 2 of higher order than #*, 



J ±m (r)=(±)»(?J)%±S . . . (25) 

 where 



2T»=*/(wi 2 -* 2 )*, ) 



n o,m iu_l i m-(m 2 -* 2 )l K26) 

 «.= -jlog2 + (m 8 -« s )i + mlog ^ KJ 



The range of variation of % w may now be investigated. 

 It was defined by 



tan X „ + !^=0. 



Now when m and z are not too nearly equal, to the second 

 order 



-— — = — m 2 /(z 2 — ra 2 )f = — sin 2 ujz cos 3 a, 



where m = £sin a. 



Thus to the same order, 



% n = sin 2 a/2* cos 3 a, (27) 



and is very small within the range of application of this 

 formula. 



