242

THEORY AND PRACTICE

TRACT 37.

atmosphere must and does decrease in density upwards, in avery rapid degree; so much so indeed, as to decrease ingeometrical progression, at altitudes which rise only in arith-metical progression ; by which it happens, that the altitudesascended are proportional only to the logarithms of the de-crease of density there. Hence it results, that the halls mustbe always less and less resisted in their ascent, with the samevelocity, and that they must consequently rise to greaterheights before they stop. It is now therefore to be consi-dered what may be the difference resulting from this circum-stance.

59. Now, the nature and measure of this decreasing den-sity, of ascents in the atmosphere, has been explained anddetermined in prop. 76, pa. 244, &c, vol. 2 of the Course.It is there shown, that if d denote the air’s density at theearth’s surface, and d its density at any altitude a, or x ; thenis x = 63551 x log. of ~ in feet, when the temperature ofthe air is 55°; we may therefore assume for the medium xz=63500 x log. ~ for a mean degree.

60. But, to get an expression for the density d, in termsof x, out of logarithms, without which it could not be intro-duced into the measure of the ball’s resistance, in a manage-able form; we find in the first place, by a neat approximateexpression for the natural number to the log. of a ratio,-j-, whose terms do not greatly differ, invented by Dr. Hal-ley, and explained in the Introduction to our Logarithms,p. 110, that X d nearly, is the number answering to

the log. I of the ratio where n denotes the modulus•43429448 &c. of the common logarithms. But, we beforefound that x — 63500 X log. of —, or is the log. of

which log. was denoted by l in the expression just above,for the number whose log. is l or — * ■; substituting there-fore for l, in the expression x d, it gives the na-