318 
[607 
607 
A MEMOIR ON PREPOTENTIALS. 
[From the Philosophical Transactions of the Royal Society of London, vol. clxv. Part ii. 
(1875), pp. 675—774. Received April 8,—Read June 10, 1875.] 
The present Memoir relates to multiple integrals expressed in terms of the (s+1) 
ultimately disappearing variables (x, .., z, w), and the same number of parameters 
(a, .;, c, e); they are of the form 
/ {(a - x)' 
+.. + (c — z) I 2 + (e — 
where p and d-ar depend only on the variables (x, .., z, w). Such an integral, in regard 
to the index 4- q, is said to be “ prepotential,” and in the particular case q = — ^ 
to be “potential.” 
I use throughout the language of hyper-tridimensional geometry: (x, .., z, w) and 
(ia, . ., c, e) are regarded as coordinates of points in (s + l)-dimensional space, the former 
of them determining the position of an element p d-cr of attracting matter, the latter 
being the attracted point; viz. we have a mass of matter —Jp distributed in such 
manner that, dix being the element of (s + 1)- or lower-dimensional volume at the point 
(x, . ., z, w), the corresponding density is p, a given function of (x, .., z, w), and that the 
element of mass pdvr exerts on the attracted point (a, .., c, e) a force inversely proportional 
to the (s + 2q + l)th power of the distance {(a — x) 2 +.. + (c — z) 2 + (e — w) 2 }b The integra 
tion is extended so as to include the whole attracting mass J p dvr; and the integral 
is then said to represent the Prepotential of the mass in regard to the point (a, .., c, e). 
In the particular case 5=2, q = — i, the force is as the inverse square of the distance, 
and the integral represents the Potential in the ordinary sense of the word. 
The element of volume dvr is usually either the element of solid (spatial or (s +1)- 
dimensional) volume dx . . dzdw, or else the element of superficial (s-dimensional) 
volume dS. In particular, when the surface (s-dimensional locus) is the (s-dimensional)