376

MISCELLANEOUS

TRACT 38.

whole weight at any place as b, the weight or quantity inthe stratum b be subtracted, the remainder is the weight atthe next stratum c; that is, from each weight subtracting apart which is proportional to itself, leaves the next weight;or, which is the same thing, from each density subtracting apart which is proportional to itself, leaves the next density.But when any quantities are continually diminished by partswhich are proportional to themselves, the remainders forma series of continued proportionals: consequently these den-sities are in geometrical progression.

Thus, if the first density be d, and from each be taken

its wth part; there will then remain its part, or the-^-part, putting m for n —1 ; and therefore the series of den-sities will be d, —d, —d, —d, —d, etc, the common ra-tio of the series being that of n to m.

SCHOLIUM.

Because the terms of an arithmetical series, are pro-portional to the logarithms of the terms of a geometricalseries: therefore different altitudes above the earth’s surface,are as the logarithms of the densities, or of the weights ofair, at those altitudes.

So that, if d denote the density at the altitude A,and d - the density at the altitude a ;then A being as the log. of d, and a as the log. of d,the dif. of alt. a — a will be as the log. d — log. d. or log. dk.

And if A —0, or d the density at the surface of the earth;then any altitude above the surface a, is as the log. of

Or, in general, the log. of -k is as the altitude of the oneplace above the other, whether the lower place be at thesurface of the earth, or any where else.

And from this property is derived the method of deter-mining the heights of mountains and other eminences, bythe barometer. For, by taking, with this instrument, the