122 



PROFESSOR STOKES, ON THE DISCONTINUITY 



when \/ — 1 OB is written for a?, occurs in many physical investigations, for example the problem 

 of annular waves in shallow water, and that of diffraction in the case of a circular disk. I had 

 occasion to employ the integral with a logarithm in determining the motion of a fluid about a 

 long cylindrical rod oscillating as a pendulum, the internal friction of the fluid itself being 

 taken into account*. In that paper the integral of (31) both in ascending and in descending 

 series was employed, but the discussion of the equation was not quite completed, one of the 

 arbitrary constants being left undetermined. A knowledge of the value of this constant was 

 not required for determining the resultant force of the fluid on the pendulum, which was the 

 great object of the investigation, but would have been required for determining the motion of 

 the fluid at a great distance from the pendulum. 



21. The three forms of the integral of (31) which we shall require are given in Arts. 28 

 and 29 of my paper on pendulums. The complete integral according to ascending series is 



M = (.4 + Z/ log a?) 1 1 + — 



J?' 



a'-'.*^ 



;^2 + 



where 



&= l-' + 2-' +3- 



2' . 4- . 



2^4^6' 



+ 1' 





■■)1 



(32) 



The series contained in this equation are convergent for all real or imaginary values of a?, 

 but the value of u determined by the equation is not unique, inasmuch as log as has an infinite 

 number of values. To pass from one of these to another comes to the same thing as changing 

 the constant A by some multiple of 27rJ5\/ — 1. If p, 9, the modulus and amplitude of w, be 

 supposed to be polar co-ordinates, and the expression (32) be made to vary continuously by 

 giving continuous variations to p and 9 without allowing the former to vanish, the value of 

 log a? will increase by 27rv — 1 in passing from any point in the positive direction once round 

 the origin so as to arrive at the starting point again. In order to render everything definite 

 we must specify the value of the logarithm which is supposed to be taken. 



The complete integral of (31) expressed by means of descending series is 



1^ .3^ l^ 3' . 5'' Y 



.4.t? 2 . 4 (4a7)' ~ 2 . 4. . 6 (4ia?)' "*" "'/ 

 1' 



1 



2.4.1? 



+ Dx-ie' |l 



1 + 



2 . 4 (4a7)- 

 1=.3« 



2.4.6 (4a?)' 



l^ 3^ 5^ 



+ ■ 



...} 



(33) 



2 . 4a? 2.4 {'ixf ' 2.4.6 {4.xf 



These series are ultimately divergent, and the constants C, D are discontinuous. It may 

 be shewn precisely as before that the values of 9 for which the constants are discontinuous are 



... - 47r, - 27r, 0, Ztt, 47r ... for C, 



... — Stt, — 7r, TT, Stt, ... for D. 



• Comb. Phil. Tram. Vol. IX. Part II. p. [38.] 



