646 



NA TURE 



[May 20, 1922 



integral of normal force, more properly normal field 

 intensity, and we have the equation 



/N^S=A47rM, 

 where M is the whole quantity of matter inclosed by 

 the surface, and k is the so-called gravitation constant, 

 the force between two unit masses at unit distance. 

 Take an example : Let the field be produced by a 

 uniform spherical shell of radius a, and describe a 

 sphere of radius R concentric with it. Consider a 

 point P on this sphere ; the field due to the shell must 

 by symmetry have the same intensity at every such 

 point as P, and the resultant intensity at P, which 

 we call F, must be at right angles to the surface; 

 thus we have for the surface integral of normal 

 force 47rR2F; the whole quantity of matter within 

 the surface if /> be the density of the shell, and da 

 the shell's thickness, is ^wpa'^da ; thus by the theorem 

 we have 



47rR2F =4Trk{4TrpaMa), 



that is, 



, ^Trpa^da 



F=k^- 



R2 



that is, the field intensity is the same as if the whole 

 mass of the shell were collected at the centre. 



The only parts of this proof which are not altogether 

 satisfying are those which depend on considerations 

 of symmetry ; but it will be tolerably clear that any 

 distribution of matter must attract a distant particle 

 after the manner stated, and no valid exception to 

 them can be taken. 



I shall return to this theorem of Gauss for a proof 

 of another proposition. No doubt it can be applied, 

 though Gauss its discoverer does not seem to have 

 done so, to establish other propositions in attraction. 

 We may prove the proposition with which we have just 

 been dealing by the following discussion, which shows 

 that the potential of a spherical shell at an external 

 point is the same as if the whole mass were collected 

 at the centre of the shell. The idea of potential was 

 given in the treatment of attractive forces set forth 

 in the " Mecanique Celeste " by Laplace : the name 

 -potential was given by Green, who made considerable 

 use of Laplace's idea. It is remarked somewhere, 

 though I cannot remember by whom, that it is 

 perhaps easier to show that the attractive force of a 

 spherical shell on an external particle is the same as 

 if the whole mass were collected at the centre than 

 to prove the same proposition for the potential. The 

 proposition for the attraction is proved in Thomson 

 and Tait's " Natural Philosophy " (a classic which, 

 like the other great treatises, nobody now has time 

 to read) by a reference to the poiiit which is the 

 inverse,^ with respect to the sphere, of the external 

 point. The proposition is proved also by direct 

 integration in the " Natural Philosophy." In a 

 paper on the historically famous problem of the 

 attraction of an ellipsoid I have shown how the 

 reference to the inverse point, in the case of the 

 sphere, may be dispensed with, and the proposi- 

 tion as to 'the force established by what is practi- 

 cally an instantaneous proof. I shall here modify 

 the method to give a proof of the theorem of the 

 potential. Use of the inverse point for the potential 

 was first made by my friend Mr. C. E. Wolff, and I 

 have here adopted his idea of dealing with the attrac- 

 tions of two elements at once, the two intercepted by 

 a small cone with its vertex at the point which 

 I call the point corresponding to the external point P. 

 This is the point A in the diagram (Fig. i) in which the 



' The idea of using the inverse point in attractions of spheres seems to be 

 due to Newton. See the " Principia," Book i., Proposition Ixxxii., in which 

 the attraction at an internal point of a spherical shell is deduced from that 

 at an external point when the law of attraction is any function of the 

 distance. In the text the law of the inverse square is alone considered. 



line CP intersects the shell so that A and P correspond 

 to one another, as do two corresponding points on 

 confocal ellipsoids. Of course the concentric spherical 

 surfaces on which P and A lie are a particular case 

 of confocal ellipsoids. 



Let the circle EAEj (centre C) be a section of the 

 shell by the paper, and P be the external point. 

 Through P describe a sphere, radius /, concentric 

 with the shell. Consider an element of area d'^ of 

 the shell at E. If k be the gravitation constant, 

 and 0- the surface density of the shell, the potential 

 at P due to the element is kffdSjr. Produce all the 

 radii to the boundary of dS to meet the concentric 

 spherical surface, and give a new element of area dS' 

 ( =dS .pia^) on the concentric surface at E'. From 

 the points of the periphery of ^S' draw lines all 

 passing through A. These lines will include a cone 

 of small solid angle w with vertex at A, meeting 

 the outer surface in the two elements iS' and dS^' 

 at E' and E/ respectively. The element dS>x set E^ 



corresponds to an element iS/ of the shell at E/ at 

 distance r-, from A, 



We have dS' = ur^lcos B, rfSi' = w^i^/cos e. 



The potential at P due to the two elements at E and 

 Ej is equal to the potential at A (the intersection of 

 CP with the shell) due to the elements ^S', cfS/ at 

 E', Ej', multiplied by the ratio a^\p. 



Thus if dN be the potential at P due to the pair 

 of elements at E', E^' we have 



/72 /1/2 



dY=^kcr 



' cos ^ ' 



ka-^o^f, 



V = ka- 



since (y + r')/cos = 2f. The potential at P produced 

 by the whole shell is thus given by 

 47ra2 



' T' 



since the whole solid angle subtended at A by the 

 external concentric sphere is ^ir. 



The proof of the theorem for the force is curiously 

 different from that for the potential. Consider only 

 a single element E in the diagram, and draw radii 

 through all the points of the periphery of the element 

 to meet the concentric surface through P ; an element 

 of this latter surface will be intercepted at E'. Let 

 dS be the area of the element at E, and d?>' that of 

 the element at E', and / the radius of the concentric 

 sphere through P, and a as before thg radius of the 

 shell. We have then 



dS- 



-JS' 



NO. 2742, VOL. 109] 



