238 On a difficulty in the Theory of 
points p’, taken according to the same law, to trace the surface of a third solid. Let 
Pp, Pp’, P'p’, be edges of three small pyramids with their other edges proceeding 
from P, P’, P", parallel, and having of course the same solid angle which we shall 
call ¢, denoting by 7, 7, 7”, their respective lengths, and by V, V’, V’", the values of 
the function V for each of them. Drawing pF perpendicular to PQ, the attraction 
of the pyramid Pp in the direction of PQ will be equal to ¢x PR; call this attrac- 
tion A, and let a be the radius of the sphere. 
Since 7" is the difference of r and 7’, we have 72+ r?—7r"?=2 rr’ =2 PRx P'Q', = 4 
and multiplying by 4 @ we find $77 +4 gr’*—$ ¢2”*=2a¢ x PR, that is V+ V'—V" 
=2aA. The same thing is true for any other three pyramids similarly related to 
each other, throughout the whole extent of the three solids which are exhausted by 
them atthe same time; and hence, if we now denote by /, V’, V", the whole values of 
the function V for the three solids, and by 4 the whole attraction of the first of them 
parallel to PQ on a point at P, we shall still have V+ V'’—V" =2aA. 
To express this general theorem in the notation of LapLacr, we have merely to 
observe that the attraction 4 is synonymous with — (7) , and that the quantity V" 
for the sphere is equal to 47a’. Substituting these values, we find 
V+ 2a(3) = — wat"; Lo 2 ft 
an exact equation, differing from the approximate one of Lapxace only in containing 
the quantity W", and totally independent of the nature of the surface or of the mag- 
nitude of the sphere ; the only things supposed being that all the lines drawn from P 
meet the surface again but once, and that no part of it passes beyond a plane through 
P at right angles to PQ. 
With respect to the limit of the quantity VY", it is obvious that if a hemisphere be 
described from P” as a centre with a radius equal to the greatest difference 6 between 
the lines Pp, P’p’, the solid Pp” will lie wholly within this hemisphere, and ‘con- 
sequently V" will be less than the value of V for the hemisphere, that is, less than { 
76°; for here all the little pyramids from the centre have the same length 8, and their 
bases are spread over the hemispherical surface ; wherefore V’=27 x }o:=7e. All 
this is independent of any thing but the suppositions just mentioned. 
If now PQ be supposed to be a spheroid of any sort, slightly differing from the 
sphere P’Q’, and such that the line PQ, perpendicular to the surface at P, passes 
nearly through the centre, than all the differences, of which é is the greatest, being of 
the first order, the quantity V", which is less than 7é, will be of the second order ; 
and therefore neglecting, as LarLace has done, the quantities of that order, we get the 
theorem in question. 
