lOi Proceedings of the Rojjal Irish Academy. 



becomes positive also, the factor e"- - 'i/2 . c~"^ of the right-hand member of 

 (33) aud of the first term of (34) is to be replaced by e"2 + i/2.e'''2, so that 

 instead of (35), we have the simpler expression 



This expression remains valid, as 2^' travels round the circle until it passes 

 into the region between A3f, A'M' ; here the argument of ih exceeds tt ; and 

 it may be seen that the factor e~"2 in (32) and in (36) is now replaced* by 

 e~"2 + ie^h^ and that (36) now becomes 



c«2 - » 1 _ e»i - "2 - ie'h + "2 . (37) 



When p passes out of this region, the factor (!""i for a similar reason has to 

 be replaced by e""i + ie"i , and, accordingly, we now recover the simpler 

 expression (36). This holds good again until ^j)' passes into the space between 

 the lines AN, A!N' \ in so doing, the argument of u^ is increased through 27r, 

 and thus the factor e"2 is changed into c"2 + ie"^h^ and (35) into 



gWg-M, _ g«i-»2 + ^g-«i-M2. (38) 



When i^ crosses A'N' , the factor 6"i is changed into e"i + tV"! from a similar 

 cause, and we thus again recover the simple expression (36), which remains 

 valid until y reaches its starting-point on the line AL. 



The final value of (36) is, however, not the same as the initial, but differs 

 from it by a change of sign; for the initial and final ^'alues of u^, and also 

 those of lit, are equal in magnitude and opposite in sign. 



Again, under the circumstances stated, the simple expression (36) is in 

 reality valid all round the contour; for the additional term in (35), (37), or 

 (38), as the case may be, is small compared with the larger of the others. 

 (It may be seen, however, that if the circumstances were such that the 

 circular contour cut the productions of the lines AN, A!M' between the 

 lines AL, AL' , it would not be legitimate in that region to omit the final 

 term of (35j; as will be shown below,t for sufficiently short waves there are 



^' The law of discontinaity in the form, of the approximate expressions for the Bessel functions 

 was conveniently stated by Stokes (" On the Discontinuity of the Arbitrary Constants that appear 

 as Multipliers of Semi- Convergent Series " ; Acta Mathematica, xxvi., 1902 ; Collected Papers, v., 

 p. 285). The substance of his statement is that of the two expressions — (1) e" multiplied by a 

 divergent series whose first term is unity, and (2) e"'* multiplied by a similar series — when the 

 argument of u increases through an even multiple of tt, (1) must be increased by 2icosr7r times (2) ; 

 and when tkrough an odd multiple, (2) must be increased by 2icosrir times (1), in order that they 

 may respectively continue to represent the same linear function of x^ilr[x) and .t^ /_,. (a;) . This 

 may be seen, in fact, fi-om equations (9), (10). 



t All. 21, p. HI. 



