TRACT 38.

PROBLEMS.

377

pressure or density, at the foot of a hill v for instance, andagain at the top of it, the difference of the logarithms of thesetwo pressures, or the logarithm of their quotient, will be asthe difference of altitude, or as the height of the hill; sup-posing the temperatures of the air to be the same at bothplaces, and the gravity of air not altered by the different dis-tances from the earth’s centre.

But as this formula expresses only the relations betweendifferent altitudes with respect to their densities, recoursemust be had to some experiment, to obtain the real altitudewhich corresponds to any given density, or the densitywhich corresponds to a given altitude. And there are va-rious experiments by which this may be done. The first,and most natural, is that which results from the known spe-cific gravity of air, with respect to the whole pressure of theatmosphere on the surface of the earth. Now, as the alti-tude a is always as log. ; assume h so as that a = hx log .-2-where h will be of one constant value for all altitudes; and todetermine that value, let a case be taken in which we knowthe altitude a corresponding to a known density d ; as forinstance, take a — 1 foot, or 1 inch, or some such small al-titude ; then, because the density d may be measured by thepressure of the atmosphere, or the uniform column of 27600feet, when the temperature is 55°; therefore 27600 feet willdenote the density d at the lower place, and 27599 the less

density d at 1 foot above it; consequently 1 =hx\og. ~ 6 ^ -which, by the nature of logarithms, is nearly=A x ~

= g 335 i nearly; and hence h — 63551 feet; which gives,for any altitude in general, this theorem, viz. a = 63551 xlog. or = 63551 x log. — feet, or 10592 x log. —fathoms ; where m is the column of mercury which is equalto the pressure or weight of the atmosphere at the bottom,and m that at the top of the altitude a ; and where m and mmay be taken in any measure, either feet or inches, 8tc.