1867.] Distribution of Matter, ^t. 485 



Y,-, at the surface. Call this function V^. It is to be remarked that the 

 value of at the surface is not the same as that of ^7^, V beino; the ex- 



di' dv ' ^ 



ternal potential, though Y^^and Y are tliere each equal to Y^. 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 po- 

 tential, be everywhere null, and therefore Y^^O. Now attribute to the 

 interior of S a function Y which is equal to Y-^ except over a narrow stra- 

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

 vanish, within which Y is made to deviate from Y^ in such a manner as 

 . dX 



to render the variation of continuous and rapid instead of abrupt. 



On applying equation (1), we see that the density is everywhere 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 (I) 



\^^'^1^cdydz, 



4. 



the integration extending over that portion. Let the portion in question be 

 that corresponding to a very small area A of the surface S ; we may sup- 

 pose it bounded laterally by the ultimately cylindrical surface generated by 

 a normal to S which travels round the perimeter of A. Taking now 

 rectangular coordinates A, r, of which the last is parallel to the normal 

 at one point of A, since Y is not changed in form by referring it to a new 

 set of rectausular axes, we have for the mass required 



I f/\= djx- dv- J 



a. «. 6^ ' 



Of the diiierential coefficients within brackets, the last alone becomes infinite 

 when the thickness of the stratum, and consequently the range of integra- 

 tion relativelv to A, becomes infinitely small. Y/e have in the limit 



dv- di' dv 



both differential coeflicients having their values belonging to the surface. 

 Hence we have ultimately for the mass 



\nn\_d\\ 

 4-V dv dv J ' 



Hence, if re be the superficial density, defined as the limit of the mass 

 corresponding to any small portion of the surface divided by the area of 

 that nortion, 



<" 



which is the known expression. 



