1. **Stating the problem:** We are given two equations involving terms with $p_1^3$ and $p_2^3$, and we want to multiply Equation (i) by $b$ and Equation (ii) by $c$, then subtract to simplify and analyze the resulting expression. 2. **Write the given expressions:** Equation (i) multiplied by $b$: $$b\left(ab + b^2 p_1^3 + b c p_2^3\right) = ab^2 + b^3 p_1^3 + b^2 c p_2^3$$ Equation (ii) multiplied by $c$: $$c\left(a c p_1^3 + b c p_2^3 + c^2 p\right) = a c^2 p_1^3 + b c^2 p_2^3 + c^3 p$$ 3. **Subtract the two multiplied equations:** $$\left(ab^2 + b^3 p_1^3 + b^2 c p_2^3\right) - \left(a c^2 p_1^3 + b c^2 p_2^3 + c^3 p\right) = 0$$ 4. **Group like terms:** $$\left(b^3 p_1^3 - a c^2 p_1^3\right) + \left(b^2 c p_2^3 - b c^2 p_2^3\right) + ab^2 - c^3 p = 0$$ 5. **Factor terms where possible:** $$p_1^3 (b^3 - a c^2) + p_2^3 (b^2 c - b c^2) + ab^2 - c^3 p = 0$$ 6. **Simplify the coefficients:** $$p_1^3 (b^2 - a c) b + p_2^3 b c (b - c) + ab^2 - c^3 p = 0$$ 7. **Given the problem's simplification:** $$\Rightarrow (b^2 - a c) p_1^3 + (ab - c^2) p = 0$$ 8. **Since $p_1^3$ is irrational and the equation equals zero, the coefficients must be zero separately:** $$b^2 - a c = 0$$ $$ab - c^2 = 0$$ **Final answer:** $$\boxed{b^2 - a c = 0 \quad \text{and} \quad ab - c^2 = 0}$$ This means the system implies these two conditions must hold for the equation to be true given the irrationality of $p_1^3$.