An introduction to the mathematical theory of attraction ...

162 Surfaces and Curves of the Second Deglee.
The potential V at an external point is given therefore
by the equation
V= 1 {log 2- log c - - e-2G cos 2}j. (14)
At an internal point the potential V is given by the
equation
- \a + b b
Hence at the boundary
y= 7Vo- - {1 + e2 cos 2O},
where c cosh f3 = a. Comparing this with the value of V at
the boundary given by (14), we get
Vo = 1I ( + log 2 - log c - 3), (15)
and for the potential V at an internal point we obtain the
equation
J-~11 î ~ -1a X2 f/2
V= M 4 + log 2 - log c - cosh-1 c - a+b) b (a+ b).
77= ~ i g c a(a+b) b(a+b)\
(16)
EXAMPLES.
Find the uniplanar potential of a homogeneous focaloidal band at an internal
point.
If U denote the required potential, M the uniplanar mass of the band, and
2a, 2b, and 2c, its axes, and focal interval, U is given by the equation
U-2Jfi+h'12 1s~- ~h- a, ab (a-b)(y2-x2)
U= 13 + log 2-log- c- cosh- b ( (- }.
-~2+og eos a2- + b2 + (a + b) (a +b2)
89. Confocal Homooids-The whole theory of the
attraction of ellipsoids has been derived by Chasles from the
properties of confocal hom~eoids, which depend chiefly on the
relations between corresponding points.
If there be two coaxal quadrics whose semi-axes are a, b, c,
and a', b', c', two points are said to correspond when their