Summary: In 2000, the notion of a subsequence of a double sequence was introduced [\textit{R. F. Patterson}, Int. J. Math. Math. Sci. 23, No. 1, 1--9 (2000; Zbl 0954.40005)]. Using this definition, a multidimensional analogue to a result from H. Steinhaus, that states that for any regular matrix \(A\), there exists a sequence of zeros and ones that is not \(A\)-summable, was proved. Additionally, an analogue of a result of R. C. Buck that states that a sequence \(x\) is convergent if and only if there exists a regular matrix \(A\) that sums every subsequence of \(x\) was presented. However, this definition imposes a restrictive condition on the entries of the double sequence that can be considered for the subsequence. In this article, we introduce a less restrictive new definition of a subsequence. We denote them by \(\beta \)-subsequences of a double sequence and show that analogues to these two fundamental theorems of summability still hold for these new subsequences.