MacCullagh’s formula: Using the notation of the previous exercise, let X i =

Y i + κ

i where κ

i are the components of a vector of length ρ > 0 in the direction ϵ.

Show that

(MacCullagh, 1855; Jeffreys & Jeffreys, 1956, Section 18.09). [The above expression gives the potential experienced at an external point (the origin) due to a unit mass or charge distributed as fX(x). The correction term is sometimes called the gravitational quadrupole (Kibble, 1985, Chapter 6).]

3.16 Show that the sum

∑

ϒ≥ϒ

∗

(−1)

ν−1

(ν − 1)! = δ (ϒ

∗

, 1)

is zero unless ϒ* = 1.