Homogeneous products of three variables

Prove that the sum of the homogeneous products of $n$ dimensions which can be formed of the letters $a,b,c$ and their powers is $$\frac{a^{n+2}(b-c)+b^{n+2}(c-a)+c^{n+2}(a-b)}{a^2(b-c)+b^2(c-a)+c^2(a-b)}.$$

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. XXIII. Partial Fractions, Examples XXIII, No. 26