The observations which have been made to determine the magnitude and figure of the earth, have not hitherto led to results completely satisfactory. They have indeed demonsrated the compression or oblateness of the terrestrial spheroid, but they have left an uncertainty as to the quantity of that compression, extending from about the one hundred and seventieth, to the three hundred and thirtieth part of the radius of the equator. Between these two quantities, the former of which is nearly double of the latter, most of the results are placed, but in such a manner that those best entitled to credit are much nearer to the least extreme than to the greatest.

Sir,

I Have examined my memorandums, concerning the observations I made on the condensed air in the air vault of the Devon Iron Works, near Alloa; and, according to your request, I now transmit you an account of them; and also of an experiment I made, when a partner and manager of these works, in order to increase the produce of blast furnaces.

The experiments described in this paper were suggested to me many years ago, when employed in studying the Geological System of the late Dr Hutton, by the following plausible objection, to which it seems liable.

Granite, porphyry, and basaltes, are supposed by Dr Hutton to have flowed in a state of perfect fusion into their present position; but their internal structure, being universally rough and stony, appears to contradict this hypothesis; for the result of the fusion of earthy substances, hitherto observed in our experiments, either is glass, or possesses, in some degree, the vitreous character.

On the 5th of August last, I announced to the Society that I had discovered soda in several varieties of the whinstone of Scotland, and also in lava from Mount Ætna, but did not describe the various experiments to which these substances had been subjected in my examination of them. In the following paper, therefore, I have the honour of laying an account of these experiments before the Society.

1. I Divide cubic equations into two varieties or species: the one, comprehending all cubic equations with three real roots; the other, all those with only one real root.

2. Let φ denote any angle whatever, and let τ = tan φ, the radius being unity: let also : then from the doctrine of angular sections we have

which being reduced to the form of an equation, is

Z3-3τZ2-3Z+τ=0.