XXII. A memoir on prepotentials

Abstract

The present Memoir relates to multiple integrals expressed in terms of the ( s + 1) ultimately disappearing variables ( x. . z, w ), ande the same number of parameters ( a. . c, e ), and being of the form ∫ ᶒ dw̄ / {( a - x ) 2 . . + ( e - w ) 2 } 1/2 s + q , where ᶒ and dw̄ depend only on the variables ( x. . z, w) . Such an integral, in regard to the index 1/2 s + q , is said to be "prepotential," and in the particular case q = –1/2 to be "potential." I use throughout the language of hyper-tridimensional geometry: ( x . . z , w ) and ( a .. c, e ) are regarded as coordinates of points in ( s + 1) dimensional space, the former of them determining the position of an element ᶒdw̄ of attracting matter, the latter being the attracted point; viz. we have a mass of matter = ∫ ᶒ dw̄ distributed in such manner that, dw̄ being the element of ( s + 1)- or lower-dimensional volume at the point ( x .. z , w ), the corresponding density is ᶒ , a given function of ( x . . z , w ), and that the element of mass ᶒ dw̄ exerts on the attracted point ( a . . c , e ) a force inversely propor­tional to the ( s + 2 q + 1)th power of the distance {( a — x 2 . + ( c — z ) 2 + ( e — w ) 2 } ½ . The integration is extended so as to include the whole attracting mass ∫ ᶒ dw̄ ; 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 s = 2, q = — ½, the force is as the inverse square of the distance, and the integral represents the Potential in the ordinary sense of the word.