238 Royal Society : — 



directions, which curve may be either expressed by some algebraical 

 or transcendental equation, or conceived as drawn liberd manu, and 

 thought of independently of any idea of algebraical expression. The 

 function V having been thus assigned to the space internal to S, the 

 equation ( 1 ) gives, according to what we have seen, the most general 

 expression for the density of the internal matter. 



There is, however, no distinction made in this between positive 

 and negative matter ; and if we wish to avoid introducing negative 

 matter we must restrict the function V for the space internal to 

 S to satisfy the imparity 



^y d*V ,^!Y^n 



daf + % 2 + dz 2 ^ 



It is easy from the general expression to show, what is already 

 known, that the matter may be distributed in an infinitely thin, and 

 consequently infinitely dense stratum over the surface S, and that 

 such a distribution is determinate. 



We know that there exists one and but one continuous function 

 applying to the space within S which satisfies the equation W=0, 

 and is equal to V at the surface. Call this function V 1 . It is to 



be remarked that the value of — r J ■ at the surface is not the same as 



dv 



that of —j—> V being the external potential, though V x and V are there 



each equal to V . The argument, it is to be observed, does not assume 

 that the two are different ; it merely avoids assuming that they are 

 the same ; the result will prove that they cannot be the same all over 

 S unless the density, and consequently the potential, be everywhere 

 null, and therefore V =0. Now, attribute to the interior of S a 

 function V which is equal to V x except over a narrow stratum adja- 

 cent to S, the thickness of which will in the end be supposed to 

 vanish, within which V is made to deviate from V, in such a manner 



as to render the variation of -j- continuous and rapid instead of 



abrupt. On applying equation (1), we see that the density is every- 

 where null except within this stratum, in which it is very great, 

 and in the limit infinite. For the total quantity of matter contained 

 in any portion of the stratum, we have from ( 1 ) 



m 



^Ydocdydz, 



the integration extending over that portion. Let the portion in ques- 

 tion be that corresponding to a very small area A of the surface S ; 

 we may suppose it bounded laterally by the ultimately cylindrical 

 surface generated by a normal to S which travels round the peri- 

 meter of A. Taking now rectangular coordinates \, fx, r, of which 

 the last is parallel to the normal at one point of A, since V is not 

 changed in form by referring it to a new set of rectangular axes, we 



