K0 = 4πβ-4, T0 = 3πβ-5. (40)
The range of the attractive force is mathematically infinite, but practically of the order β-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 φ(z) = zn. We get
| Π(z) = − | zn+1 | ψ(z) = | zn+3 | , |
| n + 1 | n + 3·n + 1 |
| K0 = | 2πzn+4 | |∞0 (41) |
| n + 4·n + 3·n + 1 |
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.
| φ(z) = 1 from z = 0 to z = a, | } (42) |
| φ(z) = 0 from z = a to z = ∞; |
and corresponding therewith,
| Π(z) = a − z from z = 0 to z = a, | } (43) |
| Π(z) = 0 from z = a to z = ∞, |