8 Prof. J. W. Nicholson on 'Zonal 



By the use o£ Binet's formula when z is large,, we readily 



find that the second term of the bracket is of order — when 



m 



m is large, so that the integral tends, for large values of m, 



7T 2 



to — . This is important in regard to the convergency of 



series of Q functions obtained as representations of given 

 functions. 



The case of functions of odd order may be noticed briefly. 

 We show without difficulty that 



= 22 



4:17 + 1 



o (2m - 2n + l) 2 (2m + 2n + 2) 2 



= _A_^f i LA 



4m 4-3 o \(2m-2n + iy (2m + 2n-r2) 2 J 

 = 4mV3{(r 2 + 3 2+ - + (2^ 2 ) + (r 2 + i + - adinL ) 



~((2m + 2) 2+ (2m + 4) 2 + '"/J 



2 | tt 2 _ _JL ^l_ 1 



4m + 3) 4 (2m +2) 2 (2m + 3) 2 ""J' 



and finally 



£[Q 2 » +1 W]V^^^ 2 + £io g r(,)}_ + . ( 22 ) 



The formulae are clearly the same whether the order of the 

 function Q be odd or even. 



Catalan gave the following expansion : — 



• -l 7r ^ 00 A -3. ... 2n— 1\ 2 fTD , . -p . . 

 sin V= 2 S o V 2.4....2n ) t^+ife-^-iW}, 



which is typical of other useful results not apparently 

 noticed. If this be differentiated, we find, since 



,i . 1 7r ^ 00 /^ , 1X /1 • 3 . ... 2n— 1\ 2 / . 



ihat yw = 2 S °° t "+ 1 )( 2.4. ...2, ) P ^>- 



