1. In his Miscellanea Analytica of the year 1762, Waring stated the identities

where sp stands for the sum of the pth powers of n quantities α, β, γ,…. If the identities be written in the uncontracted form

it is readily seen that they hold not merely for certain powers of certain quantities, but for any quantities whatever; and that, indeed, for the elements of the array

we may legitimately substitute in order the elements of the array

thus obtaining the identities

These latter were given by Binet in 1812, and are usually, but with little real justification, spoken of as “Binet's Identities.”