336 
MR. AIRY’S ACCOUNT OF THE BARTON EXPERIMENT 
as produced by a shell of matter, or as produced by the matter between two parallel 
planes of unlimited extent. 
If the extent of the planes be limited, their form being circular, let f>=wc; 
then the difference of attractions is 4'rc.<Z.|w+ 1 — &c.^ j = 47rc.c?.|l— 
nearly. If (as in the Harton experiment) mile, ^=3 miles =12c nearly, this 
23 
= 4^0. 1 — ^ nearly. Thus it appears that ^ of the effect is produced by the 
24 
matter within three miles of the pendulum-stations. It is evident therefore that it is 
not necessary to give attention to the small inequalities of the ground at any great 
distance from the pendulum-stations. 
5/. The inequalities which we have to consider are entirely at the surface, and do 
not in any case exceed in vertical measure one-tenth of the depth of the mine. They 
may therefore be considered as being actually at the surface; and, if their horizontal 
extent is not very great, each may be considered as collected at its centre of gravity. 
Its effect on the upper station will be 0 ; its effect on the lower station will be 
jXdx volume; where § is the horizontal distance of the centre of gravitv. 
{p^ + cy 
An eminence will increase the attraction upwards, and a depression will diminish it. 
But as our only object is to find the difference of attractions on the two stations, we 
may estimate the whole, with changed sign, as an effect on the upper station ; then 
an eminence will increase the attraction downwards, and a depression will dimi- 
nish it. 
There is however one depression which it is desii’able to consider in a different 
way, namely that of the sea. The depth of the sea itself is less important than the 
depression below the table-land, which continues with little change of level to the 
edge of the cliffs ; and a sufficiently accurate estimate may be formed of the measure 
of depression (including the effect of the attraction of the water) considered as uni- 
form. The line of cliff may be considered as straight. Let a be the distance from 
the pendulum-station to the straight line of cliff, measured perpendicularly to that 
line ; h the depth of tlie depression ; let x be parallel to a, and ^ parallel to the line 
of cliff. The matter dx^xx^^xh is at the distance fi‘o™ upper station, 
and therefore its vertical attraction on the lower station is 
d.b.c.lx.h) d.b.c.lx 
or 1 
[x^ -\-y^ + c^) “ [x'^ ■+• y'^) “ 
nearly. 
Integrating first with 
tl) tC • 
respect to y, from — > oo to have — — ■ 
Integrating then with respect to x, from a to oo, we have 
2d.b,c 
a 
These formulae will suffice for our purpose. It is only necessary further to remark, 
that, as the unit of measure is absolutely arbitrary, and as the numerical and gra- 
phical operations are a little facilitated by using for unit the “depth of the mine,” I 
have, in all the subsequent calculations, adopted that quantity (1256 feet) as unit 
