Manchester Memoirs, Vol. I. ( 1 906), No. 8. 21 



resemblance, I write /'(?^ + /^), when (^^0, in the form 

 i\>{ikii), and work out the value of {0(//-«)}~^ on the 

 assumption of convergence. 



I thus find — 

 {'^{iku)\--'- 

 = I - 3/^d'^" - 3(/?^ + %V^'')t;''*" + {Vi - ^k'y^'' 



+ {h' - i/iy''"' + -V-// v^" + -V-/^''t?«''•" + . . . (29) 



In this the reduction of the magnitude of the later 

 coefficients is very noteworthy. And, although a con- 

 tinuation of the series in the coefficients is desirable, 

 it would appear that under proper conditions the right- 

 hand side is rapidly convergent and that the series can 

 be properly reversed. 



To examine the necessary conditions more fully, it 

 becomes manifest that the expression on the right-hand 

 side of (29) will have a factor of the form (i - ie"") where 

 ^ is not widely different from 3//, and to which we may 

 approximate more closely. 



Thus, when we write 



we may determine a, /3 and y approximately so as to 

 satisfy the condition 



,_3^_3(^^_+3r3 + =0 



from (29), with some amount of accuracy. The results 

 will only give an approximation to the actual values, 

 by the nature of the method used. 



The equations obtained are : 



I - a + ^\a' + sVa* + ^^n^ + ^-f ^a" = 0, 



r=-(.VaV + Tfl8«'') • • • (30). 



