418 Prof. Stokes on the Communication of I'ibration 



tweeo the two equations connecting A ; B with E, F and the two 

 connecting Cj 1) with E, F is given, except as to notation, in the 

 two equations (41) of my second paper. To render the notation 

 identical with that of the former paper, it will be sufficient to 

 write A — 1> log (im) 4- B log [imr) for A + B log r, and w for imr. 

 The ('([nations referred to may be simplified by the introduction 

 ofEuler's constant 7, the value of which is -5772 15(H) &c.j since 

 it is known that 



7r-*P(i) +^4+7=0, 



r ; (a?) denoting the derivative of the function T(n). Putting 



A-Blogt»*=A', 



we have by equations (41) of the second paper referred to 



C==(27r)-i|tA'+[(loga-7)*-7r]B}. | . . . (24) 

 D=:(27r)-*{A' + (log2-y)B}, (25) 



i being written for */ — 1. It is shown in that paper that these 

 values of C, 1) hold good when the amplitude of the imaginary 

 variable a? lies between the limits and it, or that of r (supposed 



TT TT 



to be imaginary) between the limits — - and -^, but in crossing 



either of these limits one or other of the constants 0, D is 

 changed. In the investigation of the present paper r is of course 

 real/ 



We have now 



A'= A- B log im = (7- log2)B 



for the relation between A and B arising from the condition 

 that the motion is propagated outwards from the cylinder; and 

 substituting in (22), we have for the value of Nq subject to this 

 condition 



Olf expressed by means of the descending series, 



L 1_ m5r + I,»(8wir) t l 



(26) 



(87) 



1 .2 .8(8iW 



