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moires de Vlnstitut, torn, xiii.) In the preceding year I had 

 obtained the theorems just mentioned, by considering the law 

 of the inverse fourth power ; and, as well as I remember, they 

 were deduced exactly as above, by setting out from the equa- 

 tion (a). But I did not then succeed in applying the same 

 method to the case where the law of force is that of nature, 

 probably from not perceiving that, in this case, the ellipsoid 

 ought to be divided (as Poisson has divided it) into concentric 

 and similar shells. This application requires the following 

 theorem, which is easily proved : 



Supposing A' to be another ellipsoid, concentric, similar, 

 and similarly placed with A, let the right line SPP' intersect 

 it in the points p and p', respectively adjacent to P and P' ; 

 then, if the direction of that right line be conceived to vary, 

 the rectangle under Pjo and 'P'p (or under Vp' and Y'p') will 

 be to the rectangle under SP and SP' in a constant ratio. 



Denoting the constant ratio by m, and combining this 

 theorem with the formula {f), we have 



Pp X Vp _ mr 



Now let the two surfaces A and A' be supposed to approach 

 indefinitely near each other, so as to form a very thin shell, 

 then ultimately F'p will be equal to PP', and we shall have 



Pp zz py = — , 



where m is indefinitely small. Therefore, if the point S, ex- 

 ternal to the shell, be the vertex of a pyramid whose side is 

 the right line SP, and whose section, at the unit of distance 

 from the vertex, is w, the attraction of the two portions Pp and 

 P'p' of this pyramid, which form part of the shell, will be equal 

 to mru). Hence it appears, as before, on account of the symme- 

 try of the surface B round the axis of ^, that the whole attrac- 

 tion of the shell on the point S is in the direction of that axis, 

 and consequently (as was found by Poisson) in the direction 



