7 - The Theory of the Figure of the Earth carried to the Second Order of Small Quantities

Summary

INTRODUCTION

As far as I know, Airy was the first to include quantities of the second order in investigating the theory of the Earth's figure; his paper is dated 1826, and is published in Part III. of the Philosophical Transactions of the Royal Society for that year.

He gave the formula for gravity which I have obtained below (§ 6 (40)). Our results would be literatim identical but that my e is expressed by e ÷ (1 – e) in his notation, and that I denote by − f the quantity which he wrote as A. He also established equations, equivalent to my (13) and (14), which express the identity of the surfaces of equal density with the level surfaces. He remarked that these may be reduced to the form of differential equations, but he did not give the results, since he found himself unable to solve them, even for an assumed law of internal density. I have succeeded in solving these equations in this paper.

Airy further concluded that the Earth's surface must be depressed below the level of the true ellipsoid in middle latitudes. He gave no numerical estimate of this depression, but expressed the opinion that it must be very small.

In the second volume of his Höhere Geodäsie, Dr Helmert has also investigated the formula for gravity to the second order of small quantities.