504 



Mr. W. M. Hicks. On Toroidal Functions. [Mar. 3,. 



seen to be more advanced than at the part which was free from pig- 

 ment. The pigment was packed in the tnbes around and between the 

 spores ; but, by focussing, it could be seen that the substance of the 

 spore was free from it. The free spores and short rods were free from 

 pigment. 



The bacteria in which it was observed showed no other peculiarities, 

 and were of about the same calibre as the rod bacteria usually observed. 



The fact is noted as affording proof that bacteria can take up 

 minute solid particles through their walls. 



V. "On Toroidal Functions." By W. M. Hicks, M.A. St. 

 John's College, Cambridge. Communicated by J. W. L. 

 Glaisher, F.R.S. Received February 21, 1881. 



(Abstract.) 



This paper contains the development of a theory for functions which 

 satisfy Laplace's equation, and are suitable for conditions given over 

 the surface of a circular anchor ring, and which therefore seem 

 important in the possibility of their application to the theory of vortex 

 rings, as well as other physical problems. From the nature of the 

 case, it will not be easy to give an intelligent and full abstract of the 

 results without making it unduly long, but it may be possible to give 

 some idea of its scope and the method of development. 



Curvilinear co-ordinates are employed, the orthogonal surfaces being 

 those formed by the revolution of a system of circles through two 

 fixed points, and the circles orthogonal to them, whilst the third 

 system are planes through the axis of revolution. Calling these 

 v, it, w, it is shown that any potential function can be expanded in the 

 form — 



0= V cosh u— cos v 22 (AP, /Z _ » + BQ w .») cos (nv + a) cos (mw + fi) 



where P m .«, Q m . n are particular integrals of a certain differential equa- 

 tion, and which are the toroidal functions whose discussion forms the 

 principal part of the paper. They are in fact the same as spherical 

 harmonics of the first and second kinds of imaginary argument, and of 

 orders of the form (2p + l)/2. It is shown how the P can be expressed 

 in terms of the first and second complete elliptic integrals F, E" ; and 

 the Q in terms of the complementary integrals F', E'. Several in- 

 teresting results are arrived at, amongst others on relations between 

 the P and Q functions, e.g., between the zonal functions (m — v) 



P x + x Q tt — P»Q„+i=2 /(2^+l). The P n serve for expansions in the 

 space outside a tore, whilst the Q n serve for space within. 



