176 PROFESSOR STOKES, ON THE NUMERICAL CALCULATION OF A 



integral powers of n, which is convergent for all values of n, real or imaginary, however great 



be the modulus, will continue to represent u when n is imaginary. The differential equation 



(11), and consequently the descending series derived from it, will also hold good when n is 



imaginary ; but since this series contains radicals, while U is itself a rational function of n, we 



might expect beforehand that in passing from one imaginary value of n to another it should 



sometimes be necessary to change the sign of a radical, or make some equivalent change in the 



— t n\ $ 



coefficients A, B. Let n =n 1 e""^- 1 i where w, is positive. Since both values of 2 (-) are 



employed in the series, with different arbitrary constants, we may without loss of generality 

 suppose that value of r$ which has | v for its amplitude to be employed in the circular func- 

 tions or exponentials, as well as in the expression for S. In the multiplier we may always 



take - - for the amplitude of n~l by including in the constant coefficients the factor by 



which one fourth root of n differs from another ; but then we must expect to find the arbitrary 

 constants discontinuous. In fact, if we observe the forms of R and S, and suppose the 

 circular functions in (15) expanded in ascending series, it is evident that the expression for U 

 will be of the form 



An~iN+ BniJV, (25) 



where N and N' are rational functions of n. At least, this will be the case if we regard as 

 a rational function a series involving descending integral powers of n, and which is at first 

 rapidly convergent, though ultimately divergent, or rather, if we regard as such the function 

 to which the convergent part of the series is a very close approximation when the modulus of n 

 is at all large. Now, if A and B retained the same values throughout, the above expression 

 would not recur till v was increased by 8tt, whereas U recurs when v is increased by Zx. 

 If we write v + 2tt for v, and observe that N and N' recur, the expression (25) will become 



- \/~l An-lN + \/ ~-lBn*N' ; 



and since U recurs it appears that A, B become \/ - 1 A, - \/ - 1 B, respectively, when y is 



increased by 27r. Also the imaginary part of the expression (25) changes sign with y, as it 



ought ; so that, in order to know what A and B are generally, it would be sufficient to 



know what they are from y - to v tk jr. 



If we put n x e* -I for n in the second member of equation (15), and write /3 for 2 . S~$ Wl i, 



and R\, Si for what R, S become when w x is put for n in the second members of equations 



(16) and all the terms are taken positively, we shall get as our result 



— i — 

 le" ~\c l \{A - \/~\ B) (fl, + S,) e P + (A + \/~\ B) (R, - Sk) e-P\. 



Now the part of this expression which contains (i?i + S t ) e 13 ought to disappear, as appears 

 from (17). If we omit the first part of the expression, and in the second part put for A and 

 B their values given by (24), we shall obtain an expression which will be identical with the 

 second member of (17) provided 



(26) 



2.3* 



