Then, if the outside cylinder is free to move
| X1 = 0, | V1 | = | 2a2 | , X = πρa2U | b2 − a2 | . |
| U | b2 + a2 | b2 + a2 |
(22)
But if the outside cylinder is moved with velocity U1, and the inside cylinder is solid or filled with liquid of density σ,
| X = −πρa2U, | U1 | = | 2ρb2 | , |
| U | ρ (b2 + a2) + σ (b2 − a2) |
| U − U1 | = | (ρ − σ) (b2 − a2) | , |
| U1 | ρ (b2 + a2) + σ (b2 − a2) |
(23)
and the inside cylinder starts forward or backward with respect to the outside cylinder, according as ρ > or < σ.
30. The expression for ω in (1) § 29 may be increased by the addition of the term
im log z = −mθ + im log r,