Read before the New York Mathematical Society, June 3, 1893.
1. There is a type of equations that remain unaltered when x is changed into – 1/X.
From the character of the roots, and from the relation that such equations bear to those named by the text-writers "reciprocal equations," the suggestive name for this type is negative-reciprocal equations.
In order to preserve a clear distinction, "reciprocal equations" will herein be referred to as positive-reciprocal equations.
To find the conditions inherent in negative-reciprocal equations.—Let the proposed equation be substitute – 1/x for x, and divide by pn, whence we have.
2. If n be odd, the identity of equations (2) and (1) requires.
Therefore, negative-reciprocal equations of odd degree will be dismissed from further consideration, as involving imaginary coefficients. This is in contrast with the, case of positive-reciprocal equations, in which the coefficients are real whether n be odd or even.