172 PROFESSOR STOKES, ON THE NUMERICAL CALCULATION OF A 



Secondly, suppose n negative, and equal to - ri. Then, writing - ri for n in (14), 

 and changing the arbitrary constant, and the sign of the radical, we get 



„ „ , i -Vt f 1-5 1.5.7.11 1 



U=Cri-*G 3 Ml — — - + - }. . . 



[ l.l6(8n*)i i.2.i6.3« 3 J 



(17) 



It is needless to write down the part of the complete integral of (11) which involves an 

 exponential with a positive index, because, as has been already remarked, it does not appear in 

 the particular integral with which we are concerned. 



7. When n or ri is at all large, the series (16) or (17) are at first rapidly convergent, 

 but they are ultimately in all cases hypergeometrically divergent. Notwithstanding this 

 divergence, we may employ the series in numerical calculation, provided we do not take in 

 the divergent terms. The employment of the series may be justified by the following con- 

 siderations. 



Suppose that we stop after taking a finite number of terms of the series (16) or (17), the 

 terms about where we stop being so small that we may regard them as insensible ; and let U x 

 be the result so obtained. From the mode in which the constants A, B, C, ... a, /3, y... in 

 (13) were determined, it is evident that if we form the expression 



d* U x n Tr d 2 U, ri „ 



dri 3 an* 8 



according as n is positive or negative, the terms will destroy each other, except one or two at 

 the end, which remain undestroyed. These terms will be of the same order of magnitude as 

 the terms at the part of the series (16) or (17) where we stopped, and therefore will be 

 insensible for the value of n or ri for which we are calculating the series numerically, and, 

 much more, for all superior values. Suppose the arbitrary constants A, B in (16) determined 

 by means of the ultimate form of U for n = 00 , and C in (17) by means of the ultimate form 

 of U for ri = 00 . Then U\ satisfies exactly a differential equation which differs from (11) by 

 having the zero at the second side replaced by a quantity which is insensible for the value of 

 n or ri with which we are at work, and which is still smaller for values comprised between that 

 and the particular value, (namely 00 ,) by means of which the arbitrary constants were deter- 

 mined so as to make C7" a and U agree. Hence Ux will be a near approximation to U. But if 

 we went too far in the series (16) or (17), so as, after having gone through the insensible 

 terms, to take in some terms which were not insensible, the differential equation which U x 

 would satisfy exactly would differ sensibly from (11), and the value of U\ obtained would be 

 faulty. 



8. It remains to determine the arbitrary constants A, B, C. For this purpose consider 

 the integral 



Q= f*e-* + *f*dx, (18) 



where q is any imaginary quantity whose amplitude does not lie beyond the limits - — and 



