The behaviour of internal waves in vertically inhomogeneous fluids has been studied theoretically by various authors, from the point of view of general hydrodynamics as well as of meteorology and of physical oceanography (see the list of references, which, however, is not meant to be complete). For mathematical reasons most of them assumed discontinuities at certain levels, either in the density or in its first derivative with respect to the vertical coordinate, z. Any transition layer was often assumed to be thin in comparison with the wave length. FJELDSTAD [7], on the other hand, by using numerical integration, succeeded in giving an.approximate method of solving the problem for certain general density-distributions, such as may actually occur in the sea, a method, however, which is only applicable for very long waves. Besides this restriction to long waves only, FJELDSTAD's method has one other disadvantage, viz. of not directly yielding general rules or relations between the properties of the internal waves and certain parameters of the density-distribution. It is therefore, that we have gone back to somewhat more special density distributions, which are perfectly continuous with respect to the density and its first derivative to z (as are FJELDSTAD's density-distributions), but which appear to be capable of an analytical treatment; furthermore, the results are also valid for small wavelengths. In the present paper we deal only with fluids extending to infinity both upwards and downwards. At first sight this seems rather unrealistic. We know, however, that the wave-motions are always confined to a certain layer, above and below which they are negligibly small, so that, if only the boundaries of the fluid fall without this layer, they will not interfere essentially with the solutions we shall find here. The thickness of this layer depends on the wave length (see section 5). . For the rest, it is quite possible to introduce a free surface and a rigid bottom, if necessary. This will make the computations much more complicated and laborious, however.