FOR DETERMINING THE MEAN DENSITY OF THE EARTH. 
333 
upper and lower stations ; if we refer to the simple investigation in article 2, it will 
appear desirable to conduct the calculation which is to apply to the earth’s irregular 
form in such a manner as to preserve the characteristics of that simpler investigation 
as closely as possible. These characteristics are, — 
1. That a shell may be traced, whose inner surface passes through the lower 
station, and whose attraction at that lower station is =0. 
2. That the attraction of the same shell at the upper station is the same as if its 
matter were collected at the centre of the earth. 
3. That the attraction of the inclosed nucleus follows the same law, in reference 
to the difference at the upper and lower stations, as if all its matter were 
collected at the centre of the earth. 
And we are to find how nearly we can approach to these circumstances on the sup- 
position that the earth’s constitution is irregular, both in the neighbourhood of Har- 
ton and in distant regions. 
51. Now if there are sensible irregularities near the upper station, it will be impos- 
sible to satisfy the first and second conditions at the same time. For, the demon- 
stration of the evanescence of shell-attraction at the lower station rests upon this ; 
that if chords be drawn through the point L, Plate XL fig. 1, included within the 
solid angle a\J}, as cLFC, the portion FC must be equal to Lc ; and therefore, if in 
the surface ab (which may be a very minute field) there be an elevation or depres- 
sion, there must be a corresponding elevation or depression over the whole AB (which 
will be, in extent, a large continent) ; and this will disturb the second condition. It 
will be better, therefore, in the first instance, to give no attention to the local irregu- 
larities near the upper station ; to assume that the sui-face there is spherical ; to find 
with this assumption how we can satisfy the three conditions ; and afterwards to 
make allowance for the effect of the irregularities near the upper station. 
52. In fig. 2, then, conceive that for some distance on each side of L (say twenty 
or thirty times the depth of L) the external surface is sensibly spherical ; and con- 
ceive that at A, B, C, &c. there are local irregularities, perhaps large in extent as 
compared with the depth of L, but very small as compared with AB in fig. 1. Trace 
the inner surface DEF by making KD=ha, BE=L6, CF=Lc, &c. These lines, 
however, are to be made geometrically equal only when the density of the matter is 
the same as that above L ; if the density about A is less than that above L, take the 
geometrical length AD greater than La in the same proportion. Then the attraction 
of the shell on the point L will be strictly equal to 0. Moreover, its attraction on 
the point U will be sensibly the same as if its form were free from irregularities. For, 
the attractions on U are all added together; the irregularities are local and nume- 
rous, and are partly additive and partly subtractive ; and by hypothesis we have 
excluded all the irregularities near L or U, which, individually, can be important. 
And it may be accepted as a universal principle, that when a result is produced by 
the addition of a great number of small components vtdnch are liahle individually to 
