Let umn, vmn be functions of m and n, vmn being real and positive for all positive values of m and n. Suppose that either vmn increases steadily to infinity with n, or that umn both tend to zero (the latter steadily) as n → ∞, for any fixed value of m. Denote by wmn, and assume that wmn exists for every value of m, being denoted by lm. Then from Stolz' extension of a result proved by Cauchy, and an allied theorem, we have , for all values of m. It follows from Pringsheim's Theorem that if the double limit of exists, being l, then lm → l as m → ∞.