144 Properties of Newtonian Potentials at Points of Free Space.

2 . Given a distribution whose density is nowhere negative, show that if the origin of coordinates is taken at the center of mass, the development in falling powers of the distance lacks the terms of order — 1 in ж, у, z, and if, in addition, the axes are taken along the principle axes of inertia of the distribution, the initial terms of the development are

и =^4- (-g 4- g - 2 Л) ЛГ« + (С 4- Л - 2 В) yg -h (^ + В -- 2 С) гД

where Л, В, С are the moments of inertia about the axes.

3 . Show that if the development of the potential of a distribution be broken off,

JW Щ (X, y, z) H„{x, y.z)

U^j - b - - - - - ^3 - - - - - + ... + -___ 4. A,,

the remainder R^ is subject to the inequality

i ? n| : ^

T

■ - T

where a is the radius of a sphere about the origin containing all the masses, and b is the radius of a larger concentric sphere, to the exterior of which P (x, y, z) is confined.

4 . Show that at distances from the center of mass of a body, greater than ten times the radius of a sphere about the center of mass and containing the body, the equipotential surfaces vary in distance from the center of mass by less than 1.2 per cent. Show that the equipotentials of bounded distributions of Iюsitive mass approach spheres as they recede from the distribution.

8 . Behavior of Newtonian Potentials at Great Distances.

We have seen that at great distances, developments hold for the potential of bounded distributions,

Ж H,(a,y,z) dU_ _Mx H^(x,y,z)

Ô "^ Q^ '^ ' " * dx "^ ßS -r g6 -r • • •.

the termwise differentation being permitted because the resulting series is uniformly convergent. Similar expressions exist for the other partial derivatives of the first order. From these we derive the important properties of the usual potentials at great distances:

Theorem V, If U is the potential of any bounded distribution of one of the usual types, then at a great distance q from any fixed point, the quantities

rr 2^^ 2^^ 2^^

QU , e^. Q^^. Q^-^

are aU bounded. As P{x, y, z) recedes to infinity in any direction, qU approaches the total mass of the disiribidion.