The goal of this paper is to relax the conditions of the following theorem: Let $A$ be a compact closed set; let the double sequence of function $$\matrix{s_{1,1}(x), & s_{1,2}(x) &s_{1,3}(x) &\dots \\ s_{2,1}(x), &s_{2,2}(x) &s_{2,3}(x) &\dots \\ s_{3,1}(x), &s_{3,2}(x) &s_{3,3}(x) &\dots \\ \vdots & \vdots & \vdots &\ddots} $$ have the following properties: 1. for each $(m,n)s_{m,n}(x)$ is continuous in $A$; 2. for each $x$ in $A$ we have P-$\lim_{m,n}s_{m,n}(x)=s(x)$; 3. $s(x)$ is continuous in $A$; 4. there exists $M$ such that for all $(m,n)$ and all $x$ in $A|s_{m,n}(x)|\leq M$. Then there exists a $\Cal T$-transformation such that \[ \text{P}-lim_{{mn}\sigma_{mn}(x)=s(x)\text{ uniformly in }A \] and to that end we obtain the following. In order that the transformation be such that \[ \text{P}-lim_{s\to s_0(S);\,t\to t_0(T)}\sigma (s;t;x)=0 \] uniformly with respect $x$ for every double sequence of continuous functions $(s_{m,n}(x))$ define over $A$ such that $s_{m,n}(x)$ is bounded over $A$ and for all $(m,n)$ and $\text P-\lim_{m,n}s_{m,n}(x)=0$ over $A$ it is necessary and sufficient that \[ \text{P}-lim_{s\to s_0(S);\,t\to t_0(T)}\sum_{k,l=1,1}^{\infty,\infty}|a_{k,l}(s,t)|=0. \]