Classes of maximum numbers associated with two symmetric equations in $N$ reciprocals

Presented to the Society, April 22, 1955; received by the editors February 5, 1955 and, in revised form, February 9, 1956. 1 Due to ingenuity of the referee, we have changed our method of reasoning about this material; a different use of relations (6) here has caused uis to modify our transformation procedure. On account of this, we have replaced our former "E-solutions' by much more general 'admissible solutions." Consequently, we have generalized the result in our first manuscript to Theorem 1 here, and we have done so relatively briefly. Furthermore, we have identified existentially every admissible solution of equation (1) that we would use in passing from a given admissible solution x of (1), x different from the Kellogg solution w in (5), to w. The material of ?4 here would not have occurred to us; it is entirely due to the referee, and it is largely in his wording. We are extremely grateful for his assistance, so much so that we offered to share the title of this paper with him. 2 Numbers in square brackets refer to papers whose titles appear in the list of references at the end of this article.