| ε | (Φ + Θ). |
| 2π |
If the greatest value of |(U − U0)| on the greater arc AB be called H, the last component is numerically less than
| H | (a² − r²) |
| D² |
of which, when the circle, of centre P0, passing through A′B′ is sufficiently small, the factor a² − r² is arbitrarily small. Thus it appears that u′ is a function of the position of Q whose limit, when Q, interior to the original circle, approaches indefinitely near to P0, is U0. From the form
| u′ = | 1 | ∫ Udω − | 1 | ∫ Udθ, |
| π | 2π |
since the inclination of QP to a fixed direction is, when Q varies, P remaining fixed, a solution of the differential equation
| ∂²ψ | + | ∂² | = 0, |
| ∂x² | ∂y² |
where z, = x + iy, is the point Q, we infer that u′ is a differentiable function satisfying this equation; indeed, when r < a, we can write
| 1 | ∫ U | (a² − r²) | dθ = | 1 | ∫ U [ 1 + 2 | r | cos (θ − φ) + 2 | r² | cos 2(θ − φ) + ... ] dθ |
| 2π | a² + r² − 2ar cos (θ − φ) | 2π | a | a² |
= a0 + a1x + b1y + a2 (x² − y²) + 2b2xy + ...,