PROF. H. M. MACDONALD ON THE DIFFK ACTION OF 



Writing 



v o = 

 and remembering that 



it follows that 



where (l + 2w ( X )/(l 2w w' ) differs from unity by a small quantity for all values 

 of n greater than z , and T is a negative quantity increasing numerically with n ; 

 whence 



and 



43 = (27r)-''' ! (sin 0)'-' (xnT 1 \ 2 (n + $)'' RV^V 



L'o A-- '/3 



- , 



\l-2w w r j 



In the series 



writing n + ^ = z (t + ^ since 2T = log tan <, the coefficient of t, in 2T is of the 

 order 2 ~' /3 ! hence, if the value of 



where n+^ = z {) is of higher order than z " 1/3 , the sum of this double series is of lower 

 order than its first term multiplied by z v ', and therefore to the order required 

 negligible. Again, in the series 



J (n 4-D'/' R'< <R;%' 



--o-A.-,,' /3 



the factor e 2l <*+xo> oscillates and, writing 



/ (4c)- 1 (3/x) 2 

 [see Appendix (vi.)], where 



3/t = - 6H-/., c ^ 2- V/'H ( -i), 



and therefore the sum of this series is of lower order than its last term multiplied 

 by z 1/3 when the value of 



e+ <ty + tyi 

 dn dn 



