By expanding the dependent variables in series of orthogonal functions on the one hand, and expanding the coefficients of these functions in power series of a parameter η on the other hand, a solution has been obtained for the system of non-linear equations of cellular convection. The expansion parameter η is chosen in such a way as to make it remain less than 1 for all finite values of the Rayleigh number. The solution so obtained is found to be valid for a large range of the imposed temperature difference, and converges rapidly. This solution provides a quantitative theory for the convective heat transport as a function of the temperature difference in the range of laminar flow.

The solution also reveals that when the actual Rayleigh number is greater than twice the critical Rayleigh number, a layer of isothermal (adiabatic lapserate in a gas medium) mean temperature develops in the middle of the fluid layer. The thickness of this layer increases as the actual Rayleigh number increases, and at the same time the temperature gradient increases in the boundary layer so that an increase in the heat transport is accomplished.

The solution reveals further that the large temperture gradients are concentrated in the region where the cold descending current approaches the lower boundary and where the warm ascending current approaches the upper boundary. It is also shown that these ascending and descending currents spread out in mushroom-like patterns, a feature characteristic of the convection of isolated hot bubbles, but one which never has been considered as the form for finite cellular convection. Recent optical observations indicate that this is the most common form of the temperature field.

The heat transport given by this solution fits a power law of exponent 1.24, which is very close to the observed power law of exponent 1.25 for laminar flow.