607] 
A MEMOIR ON PREPOTENTIALS. 
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the potential-solid integral, provided that the point (a, .., c, e) does not lie within the 
material space: I would rather say that the integral does not satisfy the equation, 
but of this more hereafter; and it is satisfied by 
■(C), 
the potential-surface integral. The potential-plane integral (B), as a particular case of 
(C), of course also satisfies the equation. 
Each of the four cases give rise to what may be called a distribution-theorem ; 
viz. given V a function of (a, .., c, e) satisfying certain prescribed conditions, but 
otherwise arbitrary, then the form of the theorem is that there exists and that we 
can find an expression for p, 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 1835, already referred to. Theorem C, in 
the particular case of tridimensional 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 général de vérifier l’expression du potentiel relatif à 
une masse quelconque homogène ou hétérogène,” 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 
limiting condition x* + ... + z 1 — R-, the prepotential of a uniform (s-coordinal) circular 
diskj-, radius R, in regard to a point (0, .., 0, e) on the axis; and in particular the 
* [Also Crelle, t. xxxix., pp. 73—89, t. xliv., pp. 356—374, t. xlvii., pp. 161—221; Green's Mathematical 
Papers, pp. 1—115.] 
** [“ Allgemeine Lehrsätze in Beziehung auf die im verkehrten Verhältnisse des Quadrats der Entfernung 
wirkenden Anziehungs- und Abstossungskräfte,” Ges. Werke, t. v., pp. 195—242.] 
+ 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 coordinai refers to the number of coordinates which enter into the equation; 
thus, x 2 +... +« 8 + w*=/* is an (s + 1)-coordinai sphere (observe that the surface of such a sphere is s-dimensional); 
x 2 +... +z 2 =f 2 , according as we tacitly associate with it the condition zs = 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 w — 0, the 
locus may be considered as an s-coordinal sphere, its surface being (s -1)-dimensional. 
C. IX. 
41