1890.] On Stokes s Current Function. 51 



Reverting now to the expression (1), it will be seen that the direct 

 distance of any point from a point on the axis of symmetry plays 

 the same part in the theory of Stokes's current function that is played 

 by its reciprocal in the theory of the potential function belonging to 

 symmetrical distributions of matter. 



Thus if r , 0, r, 6, be the coordinates of a point upon the axis, 

 and of any other point, the distance between these points, 

 x/C^o 2 2r r cos 0+r 2 ), may be developed in a convergent series, say 



! f ~^ J - (cos 0} or ".to -** ln (cos 0) ' 



according as r is greater or less than r, I fl (cos 0) being a certain 

 function of 0, and we see from (6) that 



(1 _^) 4+rc (n-1) 1,00 = ......... (7). 



Now it is evident from the analogue of zonal harmonics that it is 

 proper to discuss the function I n (cos 0), and other solutions of (7) 

 before considering the applications of Stokes's current function to 

 the motion of liquids. It is with this discussion that the first three 

 chapters are occupied, and, as might be expected, the theory closely 

 resembles that of spherical harmonics. I have accordingly made 

 free use of the order and methods adopted by Heine in his ' Hand- 

 buch d. Kugelfunctionen,' more especially in chapters i and ii,* 

 where the necessary changes were slight. Moreover, the functions I 

 deal with have themselves been discussed by Heine, on a different 

 method, and most of the expressions which I find in the following 

 pages are given by him. Full references to these are given in 18. 



The idea of developing the solutions of D^ = in a manner more 

 or less analogous to that employed with regard to Laplace's equation 

 appears to have been first used by 0. E. Meyer, f who obtains the 

 equation (7), shows that the functions contain 1 /t 2 as a factor, and 

 that they obey (28), chapter ii. An expression which shows the 

 relation of the functions to zonal harmonics was given by Mr. 

 Butcher ;J and functions of fractional order have been used by 

 Mr. Hicks, in connexion with his researches on the theory of the 

 motion of vortex rings. The fuller account of such functions which 

 is found in the following pages may be of interest in relation to these ; 

 for example, I would refer to 63, chapter v. 



* The following sections of the first three chapters contain methods or results 

 which, so far as I am aware, are original : 12, 13, 17, 21, 25, 26, 29, 30, 32, 36, 38, 

 40, 42. The remainder of the paper is original, except where specially acknow- 

 ledged, or where a result is too well known for that to be necessary. 



t ' Crelle,' vol. 73, 1871. 



% ' London Math. Soc. Proc.,' vol. 8. See p. 143, chapter vi. 



' Phil. Trans.,' 1884, 1885. 



E 2 



