170 PROFESSOR STOKES, ON THE NUMERICAL CALCULATION OF A 



We get from (6) 



dp* 3 \ 6 6j ./„ \ 37 3V 6 6/ 



which becomes by (7) 



-—+-« = lv/ - 1 (10) 



Equating to zero the real part of the first member of this equation, we get 



-TT+ ~U=0 (11) 



dn % 3 v ' 



5. We might integrate this equation by series according to ascending powers of n, and we 

 should thus get, after determining the arbitrary constants, the series which have been already 

 mentioned. What is required at present is, to obtain for U an expression which shall be con- 

 venient when n is large. 



The form of the differential equation (11) already indicates the general form of £7 for large 

 values of n. For, suppose n large and positive, and let it receive a small increment $n. Then 



the proportionate increment of the coefficient — will be very small ; and if we regard this coeffi- 



3 



cient as constant, and Sn as variable, we shall get for the integral of (ll) 



U=Ncos{ \/(^) >in\ +N'sm\ \/(-).$n}, . . (12) 



where N, N' are regarded as constants, Sn being small, which does not prevent them from 

 being in the true integral of (ll) slowly varying functions of n. The approximate integral 

 (12) points out the existence of circular functions such as cos/(w), sin/(n) in the true 



integral ; and since \/ I — J . In must be the small increment of /(«), we get / (w) = f \/ — , 



omitting the constant, which it is unnecessary to add. When n is negative, and equal to - »', 



the same reasoning would point to the existence of exponentials with ± |^ \/ — in the index. 



Of course the exponential with a positive index will not appear in the particular integral of 

 (ll) with which we are concerned, but both exponentials would occur in the complete integral. 

 Whether n be positive or negative, we may, if we please, employ exponentials, which will be 

 real or imaginary as the case may be. 



6. Assume then to satisfy (11) 



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£7 = e 3 3 {Jn a + BnP+ Cn? +...}*, . . . (13) 



* The idea of multiplying the circular functions by a series according to descending powers of n was suggested to me by 



