»)X CERTAIX DTFFRAL'TIoX I'lIEXOiMEXA. 311 



It is to he remarked th:it when a = r, 7v beconios eqii:il to - J°(c), 

 wliore .7° denotes Ijessel'.s function of tlie Hrst kind witli index 0. 

 Tliiis the .'d)ove expression for Jv reduces to 



K 



_ / Ç- Ç' ç*' ç*^ ' \ 



Tlie expression within the hracket is the well-known form for -7°(ç). 



Tt is easily seen that the above two series f )r 7v and L convero-e 

 ra]>idly so lon^' as ^ is small ; hut when ç becomes lari^-e, it wonld be 

 advaiitau'cons to em])loy other expres.^ions fn* K and L. 



The usual process of calrulatini^- / e ^ ^"* "^ dç is to ex|)and c'' ^"' ^ 



in a Fourier series, and integrate eac-h term of tlie series sejiaralcly. 

 Thus 



1 



• A) ^ 



1 



E(piatinr;- the real and imau-imu'V ])arts to K and Tj respectively, 

 we ha^•e 



(c) A' = o..!" (,-) + 2 V (- 1)" r-- (ç 



2 « 



The f(^rm li'iven above is not r.a])idlv C(^nveru'ent. The values of 

 J"(ç) can l)e easily c:d(adate(l from the values of J°(^) and 'P{^) g'iven 

 in the tables of Hansen and Meissel u]) to certain values of the argai- 

 ment ^. \\\\\ for hig-hcr values of ç, we should have to calculate -/^"(c) 

 and 'J\:). ^loreover, when n exceeds ç, the value of '/"(ç) deduced 

 successively from J°(^) and -/^(ç) becomes inaccnrate, and we are thns 

 compelled to undertake the calculation se])arat('ly. Tliese considéra- 



