64 Mr> By M. Macdonald. The Bending of [May 12, 



where T 7? , T, . . . denote the values of T M (?) f T n '(^), , when 

 From the expression for R n (^) it follows that 



JR'(*) - ^ 2^ ^ _ s) n(s _ j) n( - 



hence tan x is positive and increases with n t thus x lies between 

 and JTT and 9x/i/9^ is positive ; therefore $! and S.-, both vanish. 

 When z - (n + 1) is of higher order than ^ 



hence 

 and 



therefore as ?i increases, subject -to the limiUition that ~ - (n + J) is of 

 higher order than .:'-, &(*) tends to become of the order (w + ^)>, 

 R,/(^) f tne or( i er unifc y> an d 3x/^ f tne order (w + J)-*. \\Tien 

 .:- (w4-J) is of lower order : than .r*,'it follows from the relation 



that R,i(^) is of the order (w + -J)j, R' n (:) of the order unity, and 9x>/9 

 of the order (n + J)-3. These latter conditions also hold when n + J - s 

 is of lower order than :'. When ?/ + -- is of higher order than s*, 

 R (:) is given by 



R,,(*) = 2T, 4 ( 5 ) cosh 2r M , 

 and 



R'(*) P: 2T'() cosh 2r u * 4T,, (s) sinh 2r n -~ n , 

 that is 



R-(*) = 2T n / 

 whence 





* These results are given by Loremr, . c CEurres Scientifiqiies,' vol. 1, pp. 435 

 479. 



