Althought the linear conduction of heat in plain walls by using the Laplace transformation are given numerous examples in "The conduction of heat in solids" by H.S. Carslaw and J.C. Jaeger, there are no reports bearing on the general solution of unsteady conduction of heat in multilayer plain walls, so far as the writer is aware, In the present research, which deals with a general solution for unsteady conduction of hert in multi-layer plain walls under different conditions, consists of two parts, of which the present one is the first and describes a method for obtaining the boundary temperatures and heat flows in the Laplace transformation. The second report will treat the cases when the heat production or initial temperature in any layer are considered and also by the convers transformation, form these functions temperatures and heat flows would be obtained in the functions of the time (t). Defining h (s) and q (s) as the Laplace transforms of the temperature θ (t) and heat flow Q (t) respectively, we get; [numerical formula] [numerical formula] where h_1 (s) and h_2 (s) are Laplace transforms of temperatures at both layer boundaries. From these equations, θ (t) and Q (t) will be obtained by convers transformation. In multi-layer plain walls (initial temperature is O), we get the relations of the laplace transforms of three boundary temperatures by considering the balance of heat flow at each boundary. For example, at boundary (1,2), we get; [numerical formula] where _1ω_2 (s) is the Laplace transform of heat flow produced at boundary (1,2); M_1, N_1 etc. are functions which are made up of constants of the layer shown by each suphix; _0h_1 (s), _1h_2 (s) and _2h_3 (s) are Laplace transforms of temperatures at boundaries (0, 1), (1, 2) and (2, 3) respectively. As we may obtain such a relation at each boundary, the Laplace transforms of unknown boundary temperatures would be obtained by solving simultaneous equations. For example, if we know _0h_1 (s) and _nh_i (s) which are the Laplace transforms of temperature at the boundaries (o,1) and (n,i) in the plain wall composed of n layers, the Laplane transform of temperature at boundary (ν-1, ν) would be [numerical formula] where _1A_n' (s) etc. are functions relating to the constants of the layers and composition of the multilayer wall, and are obtained by the next two recurence formulae, [numerical formula] where we define [numerical formula] Temperature and heat flow in any layer are obtained by solving (1) and (2) simultaneously, In the cases when fluid layers ar/and a semi-infinite layer are contained in the wall, and when heat flow at a boundary is shut out, we get similar results by employing _1A'_n (s) functions which are extentions of their definitions.