An analytical method is developed which yields key values of a harmonic function U in rectangular figures in terms of certain simple combinations of known boundary values, which we call Laplacian perimeters and denote by P. From the key values all the remaining interior values can be directly and rapidly calculated. The basic ideas consist of the introduction of four elementary figures of variable lengths: 3(2+δ), 3(3+δ), 4(2+δ), and 4(3+δ), where 0≤δ≤1, for which the key values are found algebraically, Eqs. (9.1) to (9.8) inclusive, and of combining analytically these elementary cases to obtain continuous solutions for rectangles (3×n′) and (4×n′) for any number n′ greater than 2. The general expression for any key value (ui)m×n′ in a rectangle (m×n′) is (ui)m×n′=C1P1+C2P2+…CiPi+…Ckpk,in which C1, C2…Ck are constant coefficients. Analytical relations are established between the coefficients for the last key value uk in a rectangle (m×n), where n is an integer, and the coefficients for the next key value uk+1 in the rectangle m(n+δ) or m(n+δ+1). Specific values of the coefficients are presented in tabular form for rectangles (3×n′) and (4×n′) with the aid of which all key values in such rectangles can be rapidly calculated. The tables also permit the calculation of any one interior key value by itself. A numerical example and further approximate formulas are given, and applications to composite rectangular areas such as angles, channels, etc., are discussed.