162 Sir William Thomson on the 



any other distribution of normal velocity than either of those, 

 the corrugation will, as it were, split into two sets of waves 



travelling with the two different velocities flo> f 1± ^— j. 



The case ?=0 is clearly exceptional, and can present no 

 undulations travelling round the cylinder. It will be consi- 

 dered later. 



The case i=l is particularly important and interesting. To 

 evaluate N for it, remark that 



I 1 (mr) = V (mr) 1 ^ fi 



and U^) = ¥o(mr)J V ^ 



Now the general solution of (24) is 



where E and D are constants. Hence, according to our nota- 

 tion, 



Io(»»-) = l + ^+p^+&c. ) . . . (37) 



the constant factor being taken so as to make I (0) = 1. 



Stokes* investigated the relation between E and D to make 

 w = when r=co , and found it to be 



E/D = log8 + 7r-*r / i=+2-079442-l-963510 = -11593n 

 or, to 20 places, > ^ 



E/D = -11593 15156 58412 44881. J 



Hence, and by convenient assumption for constant factor, 



+ '^r (Si + '11593) + ^- 2 (S a + '11593) + &c. ' 



It is to be remarked that the series in (36) and (39) are 

 convergent, however great be mr ; though for values of mr 



* u On the Effect of Internal Friction on the Motion of Pendulums," 

 equations (93) and (106). (Camb. Phil. Trans. Dec. 1850.) 



P.S. — I am informed by Mr. J. W. L. Glaisher that Gauss, in section 32 



of his " Disquisitiones Generates circa seriem infinitam l+iil£#+ &c," 



1 . y 

 {Opera, vol. hi. p. 155), gives the value of —v"^T% or -^( — 5), in his 



