Planetary Attraction and Satellite Revolutions

The attraction of a planet on its satellites varies directly as the mass ($M$) of the planet, and inversely as the square of the distance ($D$); also the square of a satellite's time of revolution varies directly as the distance and inversely as the force of attraction. If $m_1,d_1,t_1$, and $m_2,d_2,t_2$, are simultaneous values of $M,D,T$ respectively, prove that $\frac{m_1{t}_1^2}{m_2{t}_2^2}=\frac{d_1^3}{d_2^3}$. Hence find the time of revolution of that moon of Jupiter whose distance is to the distance of our Moon as $35:31$, having given that the mass of Jupiter is 343 times that of the Earth, and that the Moon's period is 27.32 days.

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. III. Variation, Examples III, No. 21