An analytical method is developed which yields key values of a harmonic functionUin rectangular figures in terms of certain simple combinations of known boundary values, which we call Laplacian perimeters and denote byP. 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+&dgr;), 3(3+&dgr;), 4(2+&dgr;), and 4(3+&dgr;), where 0≤&dgr;≤1, for which the key values are found algebraically, Eqs. (9.1) to (9.8) inclusive, and of combining analytically these elementary cases to obtaincontinuous solutionsfor rectangles (3×n′) and (4×n′) for any numbern′ 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 whichC1,C2…Ckare constant coefficients. Analytical relations are established between the coefficients for the last key valueukin a rectangle (m×n), wherenis an integer, and the coefficients for the next key valueuk+1in the rectanglem(n+&dgr;) orm(n+&dgr;+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.