Step 1: Identify the positions of each atom in the unit cell.
Given: A atoms are at body centre positions, B atoms are at corner positions, and C atoms are at face centre positions. We need to count how many of each atom effectively belong to one unit cell.
Step 2: Count the contribution of B atoms (at corners).
In a cubic unit cell, there are 8 corners. Each corner atom is shared among 8 adjacent unit cells, so each corner contributes \(\frac{1}{8}\) of an atom to one unit cell: \[ \text{B atoms per unit cell} = 8 \times \frac{1}{8} = 1 \]
Step 3: Count the contribution of A atoms (at body centre).
The body centre is entirely inside the unit cell. It is NOT shared with any other unit cell. There is 1 body centre per unit cell: \[ \text{A atoms per unit cell} = 1 \times 1 = 1 \]
Step 4: Count the contribution of C atoms (at face centres).
In a cubic unit cell, there are 6 faces. Each face-centred atom is shared between 2 adjacent unit cells, so each face-centre contributes \(\frac{1}{2}\): \[ \text{C atoms per unit cell} = 6 \times \frac{1}{2} = 3 \]
Step 5: Write the empirical formula of the compound.
We have A : B : C = 1 : 1 : 3. The simplest whole-number ratio gives the formula ABC3. This type of structure is called the perovskite structure (e.g., CaTiO3 where Ca is at body centre, Ti at corners, and O at face centres).
Step 6: State the final answer.
The formula of the compound based on the given unit cell arrangement is ABC3.
\[ \boxed{ABC_3} \]