1893.] connected with Bessel's Functions. 125 



It is proved in Lord Rayleigh's Sound, Vol. ii. p. 273, that 



WJ ^ ""1 ~ 1 . 8tr ■^2!(8«f 3! (Sir/ "^ 



= __ (^ + log i,r) /oCO - - {^A-^2S,+ ^,1,qA -)--0-'^1 



where <y is Euler's constant, and 8^^ = 1 + 2~^ + ...n~^. The 

 imaginary part of the right-hand side is obviously equal to 

 — t Jo (^). and we must now show that the integral (9) is equal to 

 the real part of either side of the above equation. This we shall 

 do by showing that the integral (8) can be expressed by means of 

 the series on the left-hand side of (11). We shall also establish 

 the legitimacy of equation (10), which was deduced from (9) by 

 differentiation with respect to r. 



In (8) put X = 1 +y and it becomes 



2 r e-^--^ryd y ^ V2e-- p ji _ i /^^' 



^...(-ThJI) ...\dy, 



where H„ 



1.3...2W-1 



2.4...2?i • 



Jo ^ "^ (2tr)" (ir)i' 



so that the integral 



irTr) '"'i^ "8^ + 21(8;^^" •••} ^^^^- 



The latter series is therefore equal to Yq (r) — iJ(, (r) ; and by 

 realizing and equating the real and imaginary parts we shall 

 obtain the two series for Yq and J^ which are given in Rayleigh's 

 Sound, Vol. II. p. 275 and Vol. i. p. 264. 



Equations (10) and also (6) may be deduced by differentiating 

 the series (12) with respect to r and then summing the two result- 

 ing series to which the differentiation leads. 



3. The circumstance that two distinct forms of the J functions 

 exist in one of which the limits are 1 and 0, whilst in the other 

 they are oo and 1, for equation (4) is equivalent to 



'"^ cos rxdx f^ sin rxda; 



(1-^^)^ J, {af-l)i' 



