The range of the attractive force is mathematically infinite, but
practically of the order [beta]^(-1), and we see that T is of higher
order in this small quantity than K. That K is in all cases of the
fourth order and T of the fifth order in the range of the forces is
obvious from (37) without integration.
An apparently simple example would be to suppose [phi](z) = z^n. We
get
z^(n+1) z^(n+3)
[PI](z) = - -------, [psi](z) = ---------,
n+1 n+3 . n+1
2[pi]z^(n+4) |[oo]
K0 = --------------- | (41)
n+4 . n+3 . n+1 |0
The intrinsic pressure will thus be infinite whatever n may be. If n +
4 be positive, the attraction of infinitely distant parts contributes
to the result; while if n + 4 be negative, the parts in immediate
contiguity act with infinite power. For the transition case, discussed
by William Sutherland (_Phil. Mag._ xxiv. p. 113, 1887), of n + 4 = 0,
K0 is also infinite. It seems therefore that nothing satisfactory can
be arrived at under this head.
As a third example, we will take the law proposed by Young, viz.
[phi](z) = 1 from z = 0 to z = a, \
[phi](z) = 0 from z = a to z = [oo]; / (42)
and corresponding therewith,
[Pi](z) = a - z from z = 0 to z = a, \
[Pi](z) = 0 from z = a to z = [oo], / (43)
[psi](z) = 1/2a(a^2 - z^2) = 1/3(a^3 - z^3) from z = 0 to z = a, \
[psi](z) = 0 from z = a to z = [oo], / (44)
Equations (37) now give
_
2[pi] / [oo] [pi]a^4
K0 = ----- | z^3dz = -------, (45)
3 _/0 6
_
[pi] / a [pi]a^5
T0 = ---- | z^4 dz = -------. (46)
8 _/0 40
The numerical results differ from those of Young, who finds that "_the
contractile force is one-third of the whole cohesive force of a
stratum of particles, equal in thickness to the interval to which the
primitive equable cohesion extends_," viz. T = (1/3)aK; whereas
according to the above calculation T = (3/20)aK. The discrepancy seems
to depend upon Young having treated the attractive force as operative
in one direction only. For further calculations on Laplace's
principles, see Rayleigh, _Phil. Mag._, Oct. Dec. 1890, or _Scientific
Papers_, vol. iii. p. 397.]