This paper discusses known mathematical equalities and inequalities found within life tables and proceeds to identify two new inequalities. The first (theorem 1) is that at any given age x, the sum of mean years lived and mean years remaining exceeds life expectancy at birth when age is greater than zero and less than the maximum lifespan. The second inequality (theorem 2) applies to the entire population and shows that the sum of mean years lived and mean years remaining exceeds life expectancy at birth. Illustrations of the two inequalities are provided as well as a discussion.

中文翻译：

---

论生命表中的数学等式和不等式：旧的和新的

本文讨论了生命表中已知的数学等式和不等式，并着手确定两个新的不等式。第一个（定理 1）是在任何给定的年龄 x，当年龄大于零且小于最大寿命时，平均寿命和平均剩余年数的总和超过出生时的预期寿命。第二个不等式（定理 2）适用于整个人口，表明平均寿命和平均剩余年数之和超过出生时的预期寿命。提供了两种不等式的说明以及讨论。