This paper considers numerical methods for the solution of Poisson's equation in an arbitrary two‐dimensional region with given boundary values. The methods are similar to those discussed by Shortley and Weller for Laplace's equation. It is shown that the entire theory of networks may be applied to Poisson's equation without change. Formulas for the treatment of irregular points and for blocks of points are developed. In addition to the mathematical considerations, the application of the methods to the case of torsion in shafts of any cross section is described. Additional formulas are derived for the determination of the maximum shear at any point in a shaft as well as the torsional stiffness of such a shaft.

1.
G. H.
Shortley
 and
R.
Weller
,
J. App. Phys.
9
,
344
(
1938
);
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G. H.
Shortley
 and
R.
Weller
,
J. App. Mech.
6
,
A
‐
71
(June,
1939
).
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2.
S. Timoshenko, Theory of Elasticity (McGraw‐Hill).
3.
If we express the value (Δψ)0 of the Laplacian at the origin in terms of the values at ψ0, ψ1, ψ2, ⋯ of ψ at (0,0), (x1,y1), (x2,y2), ⋯ by the equation
(Δψ)0 = aψ0+ ∑ i = 1⋯biψi
, we obtain the following equations for the determination of the coefficients a and bi: 
a+∑bi = 0
;
∑bixi = 0, ∑biyi = 0
;
∑bixi2 = 2, ∑biyi2 = 2, ∑bixiyi = 0
;
∑bixinyim = 0,ifn+m>2
. As many as possible of these equations, in order, are to be satisfied.
4.
If one of the corner points, n, happens to be missing because of the incidence of a boundary one can use the two d points instead, with a formula of the type
b1′ = 18[2a1′+e1+e2+2d1+2d2+3αb1h2]
. The same formula is available for the b point of the four‐block (fig. 3).
5.
Let δi = ψi′−ψi, δi′ = ψi″−ψi′; etc. To demonstrate that δ′ is derived from δ in the same way as ψ′ from ψ but with the α’s and the boundary values set equal to zero, we may proceed as follows: Let there be N interior points, improved in the order 1,2,⋯,N. Then the formula for the improvement of the kth point may be written as
ψk′ =  ∑ i = 1k−1Vkiψi′+ ∑ i = k+1NWkiψi+ ∑ b.p.Xkβψβ+Ykαk
, (A) where the V, W, X, Y, and α are constants independent of Ψ; the Ψβ are the fixed values of Ψ at the boundary points, and in general the coefficients of all but five or six terms vanish. Then
ψk″ =  ∑ i = 1k−1Vkiψi″+ ∑ i = k+1NWkiψi′+ ∑ b.p.Xkβψβ+Ykαk
;
(ψk″−ψk′) =  ∑ i = 1k−1Vki(ψi″−ψi′)+ ∑ i = k+1NWki(ψi′−ψi)
or
δk′ =  ∑ i = 1k−1Vkiδi′+ ∑ i = k+1NWkiδi
. (B) Comparison of (A) and (B) completes the demonstration.
6.
But note that the direction of the shearing force is perpendicular to the gradient of ψ. For an ordinary interior point (x, y), the two components of this gradient may be computed, to second‐order accuracy, by the formulas
gradxψ(x,y) = [ψ(x+h,y)−ψ(x−h,y)]/2h
,
gradyψ(x,y) = [ψ(x,y+h)−ψ(x,y−h)]/2h
. These are proportional to the components −τyz,τxzof the shearing stress; the maximum stress is the square root of the sum of the squares of these quantities.
7.
Reference 2, p. 248.
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© 1940 The American Institute of Physics.
1940
The American Institute of Physics
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