 Axial and transverse gravity of a uniform rod

Integrate Newtonian point-force elements along a uniform thin rod of total mass $M$ and half-length $a$. The first title formula applies on the axis outside the endpoint, $b>a$; the second applies on its perpendicular bisector. Both approach the point-mass result at large distance, but at equal distance their ratio is $1+3a^2/(2b^2)+O(a^4/b^4)$. A nonuniform linear mass profile cannot be replaced by these formulas using only total mass and length.