500 
PROFESSOR A. SCHUSTER ON THE DIURNAL 
may get as low as 4 X 10®. A whole sphere made up of such water would very 
considerably reduce the amplitude of the observed vertical force, and alter the 
resultant phase by 45° nearly. The average conductivity of the Earth, as seen from 
these examples, must he small, although it may be considerable over limited areas. 
Such limited areas would principally affect the harmonic terms of higher degrees, and 
we should not consequently expect for them such a good agreement between theory 
and observation. Table XXII. shows that as the resistance of the snhere diminishes 
X 
the retardation of the resultant phase seems to approach a constant value of 45°. 
This can be proved to be quite generally the case. It follows directly from a formula 
given by Professor Lamb in the Appendix, and can also be seen as follows :—When 
the conductivity is good the angle between OHo and OHj^ in fig. 11 will steadily 
diminish, and ultimately vanish. OK will ultimately coincide with OH^, and OR 
with OV;^; but the angle between OV 3 and OV;^ will increase towards 180°, and the 
sides will tend towards equality. Two very nearly equal and very nearly opposite 
forces may have a resultant which is inclined by a finite angle to the forces. To find 
the angle in the limit between OV^ and OV 3 we deduce, in the first place, an expres¬ 
sion for the ratio of the vertical force due to the outside effect to the vertical force 
due to induced currents for good conductivities. 
This ratio depends, as shown by Professor Lamb, on the function 
(fcR) where /c“ = 
47rpi 
and 
If n is odd, 
d \« 
Ycix) 
sinh ^ = 
cosh^ n.n -H 1 siuli^ n — l.n.oi -f l.% + 2 cosh ^ 
n +1 
2 
n + 
2 "f* 
! 4: ^ 
{n —2) .. . (n + 3) sinh f 
2.4.6 
If 71 is even, we must interchange sinh ^ and cosh 
If C is larger, so that e~^ can be neglected, compared to e^, cosh { = sinh ^ = e^, and 
OT>o 
2n + l.{2n + 3) (/cE) 
which is the ratio of the vertical forces due to the induced and inducing potential 
will become 
n + l.n + 2 
n.71 + 1.71 -t- 2.71 + 3 
.. . /o./o T J-.IO T ^.Ity -\ ^ P— ^ 
j 4 "T 2 4 ^ ‘ ‘ 
1 — 
71 — 1.71 
— 2.71 — 1.71.71 -f- 1 
... 
2.4 
