The older theories of turbulence have to treat various types of fully developed turbulence separately. For example, the ideas used in the usual momentum transfer theory cannot be applied to isotropic turbulence, where the mean flow has no velocity gradient. The purpose of the present paper is to establish differential relations for the turbulent strength and thus obtain a formal system to include all types of fully developed turbulence. \par The two fundamental equations developed are the energy equation $$ \frac{DE}{Dt}=-c\frac{E\surd E}{l}+kl\surd E\left(\frac{dU}{dy}\right)^2+\frac\partial{\partial y}\left(k_1l\surd E\frac{\partial E}{\partial y}\right), $$ and the familiar formula for Reynolds sheer, $$ \tau'=\epsilon dU/dy=k_s\surd EdU/dy; $$ where $E$ is the kinetic energy of turbulent fluctuation per unit volume, $D/Dt$ denotes substantial differentiation following the mean motion $l$ is the mixture length, $c,k$ and $k_1$ are constants, $U$ is the mean velocity in the direction of the $x$-axis, $\tau'$ is the Reynolds stress (divided by density), and $\epsilon$ is the exchange coefficient. It is shown how existing formulae can be derived from these as special cases. Wieghardt made an estimate of $c$ from the decay of isotropic turbulence and obtained a value of 0.17--0.21. For flow through a channel, satisfactory agreement with experiments is obtained, especially for distribution of turbulence strength, with $c=0.18$, $k=0.56$, $k_1=0.38$. It is noted that $c=k^3$, in agreement with the general deductions of Prandtl in the first part of the paper.