678 
PROFESSOR CATLET ON PREPOTENTIALS. 
sion for g>, the density or distribution of matter over the space or surface to which the 
theorem relates, such that the corresponding integral V has its given value, viz. in 
A and B there exists such a distribution over the plane w=0, in C such a distribution 
over a given surface, and in D such a distribution in space. The establishment, and 
exhibition in connexion with each other, of these four distribution-theorems is the 
principal object of the present memoir ; but the memoir contains other investigations 
which have presented themselves to me in treating the question. It is to be noticed 
that the theorem A belongs to Green, being in fact the fundamental theorem of his 
memoir of 1885, already referred to. Theorem C, in the particular case of tridimen- 
sional space, belongs also to him, being given in his ‘Essay on the Application of 
Mathematical Analysis to the theories of Electricity and Magnetism’ (Nottingham, 1828), 
being partially rediscovered by Gauss in the year 1840 ; and theorem D, in the same 
case of tridimensional space, to Lejeune-Dirichlet : see his memoir “ Sur un moyen 
general de verifier l’expression du potentiel relatif a une masse quelconque homogene 
ou heterogene,” Crelle , t. xxxii. pp. 80-84 (1840). I refer more particularly to these 
and other researches by Gauss, Jacobi, and others in an Annex to the present memoir. 
On the Prepotential Surface-integral. — Art. Nos. 1 to 18. 
1. In what immediately follows we require 
dx . . .dz 
(x 2 . . , + x 2 + d 
,2\*s+?’ 
limiting condition x 1 . . R 2 , the prepotential of a uniform (s-coordinal) circular 
disk*, radius It, in regard to a point (0 ... 0, e) on the axis ; and in particular the value 
is required in the case where the distance e (taken to be always positive) is indefinitely 
small in regard to the radius R. 
Writing x—r% . . . z=r%, where the s new variables £ . . . £ are such that £ 2 . . .+£ 2 — 1, 
the integral becomes 
r r*-'drdS r r R r°~ l dr 
J (r 2 + e 2 f +q ' ~J „ (r 2 + e 2 f + ^ 
where dS is the element of surface of the s-dimensional unit-sphere £ 2 . . .+£ 2 =1 ; the 
2 (IU) S 
integral J dS denotes the entire surface of this sphere, which (see Annex I.) is — ~plg - 
The other factor, 
r R r s ~ ] dr 
Jo + 
is the r-integral of Annex II. 
* It is to be throughout borne in mind that x . . . z denotes a set of s coordinates, x . . . z, w a set of s+ 1 
coordinates ; the adjective coordinal refers to the number of coordinates 'which enter into the equation ; thus, 
x 2 . . . -\-z 2 -\-w 2 —f 2 is an (s+l)coordinal sphere (observe that the surface of such a sphere is s-dimensional ) ; 
x 2 . . . +z 2 =/ 2 , according as we tacitly associate with it the condition w— 0, or w arbitrary, is an s-coordinal 
circle, or cylinder, the surface of such circle or cylinder being s-dimensional, but the circumference of the circle 
(s— l)dimensional; or if we attend only to the s-dimensional space constituted by the plane iv=Q, the locus 
may be considered as an s-coordinal sphere, its surface being (s— l)dimensional. 
