330

MISCELLANEOUS

TRACT 38.

vertical vibrations each in 1 second of time, or is isochro-nous with a common pendulum of 39f inches long, the ex-tent of each vibration of the former being 6^ inches.

PROBLEM IX.

Required to determine the quantity of matter in a sphere,the density varying as the nth power of the distance from thecentre?

Let r denote the radius of the sphere, d the density at itssurface, a — 3 - 1416 the area of a circle whose radius is 1,and x any distance from the centre. Then 4ax* will be thesurface of a sphere whose radius is x, which may be consi-dered by expansion as generating the magnitude of the solid;therefore 4ax z x will be the fluxion of the magnitude; but

as : x ”:: d : the density at the distance x, therefore

dx n 4-ndv n '^^T

4ax*x x -= — -- = the fluxion of the mass, the fluent

of which y rl —— , when x — r, is the quantity of the

matter in the whole sphere.

Corol. 1. The magnitude of a sphere whose radius is r,being ^ ar 3 , which call m; then the mass or solid content w r illbe — x m, and the mean density is

Corol. 2. It having been computed, from actual experi-ments, that the medium density of the whole mass of theearth is about 5 times the density d at the surface, we cannow determine what is the exponent of the decreasing ratioof the density from the centre to the circumference, sup-posing it to decrease by a regular law, viz, as r" ; for then itwill be Sd = and hence n = — f. So that, in this

_ 12 1 ,

case, the law of decrease is as r T , or as—, that is, in-

r J

versely as the ?/tbs power of the radius.