In this paper there is established an asymptotic formula for the number of simultaneous representations of two numbers as sums of an increasing number of terms involving a power function, i.e., an asymptotic (as $n\to\infty$) formula is found for the number of solutions in integers $x_i$, $0\le x_i\le p$, of the following system of diophantine equations: $$ \begin{cases} x_1+x_2+\dots+x_n=N_1,\\ x_1^2+x_2^2+\dots+x_n^2=N_2. \end{cases} $$ The analysis is carried out as in the proof of a local limit theorem of probability theory and involves estimates of Weyl sums.