in a series of the attraction of a spheroid. 107 
but similar to it, in a memoir read to the Academy of Sci- 
ences in 1818 ; and a third by M. Poisson, in an interesting 
and profound memoir on the distribution of heat in solid 
bodies. The two last demonstrations are fundamentally the 
same ; but as M. Poisson has stated the reasoning more 
fully, and fixed the sense of the proof more precisely, I wish 
to refer to his memoir. One observation it is proper to make, 
which is, that in the integration by which the differential 
equation is proved, the function expressing the thickness of 
the molecule is considered as invariable, or is treated as a 
constant quantity. It is essential to attend to this remark, 
which in reality affords the clue necessary to unravel what 
is mysterious in this investigation. 
In order to acquire a distinct notion of the meaning of the 
differential equation in the sense in which it is demonstrated, 
conceive the surface of the earth, perfectly smooth and spheri- 
cal, to be covered with circles, we shall say, of a thousand 
yards radius each. The circles may either touch one ano- 
ther and cover the whole surface of the earth ; or they may 
cover it partially only, and with any interruptions that can 
be imagined : conceive also that a mass of matter, or mole- 
cule, is placed within every circle ; the thicknesses of the 
molecules being entirely arbitrary, and subject to no law of 
variation or restriction whatever, excepting that they are 
quantities of inconsiderable magnitude when compared with 
the radius of the sphere. These things being supposed, the 
differential equation will be separately true of every one of 
the molecules. 
Let now the whole surface of the earth, or any portion of 
it, be covered with molecules, the thickness varying according 
