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moires de I’ Institut, tom, 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 (az). 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 Pp and P’p (or under Pp’ and P’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 P’p _ mr 
a 2 nh 
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 P’p will be equal to PP’, and we shall have 
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 mrw. 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 
