Given a $999 \times 999$ square table, our goal is to maximize the number of triples $(C_1, C_2, C_3)$ such that:
- $C_1$ and $C_3$ are white cells,
- $C_2$ is a red cell,
- $C_1$ and $C_2$ are in the same row,
- $C_2$ and $C_3$ are in the same column.

Let $w$ represent the number of white cells and $r$ the number of red cells, where $w + r = 999^2$.

To form a valid triple $(C_1, C_2, C_3)$, for each red cell $C_2$, we can choose $C_1$ from the remaining white cells in its row and $C_3$ from the remaining white cells in its column.

### Approach

1. Determine combinations for a fixed red cell:
- For each row, let there be $w_i$ white cells and $r_i$ red cells. Therefore, the number of ways to choose a pair $(C_1, C_2)$ in the same row is $r_i \cdot (w_i - 1)$.
- Likewise, for each column with $w_j$ white cells and $r_j$ red cells, the number of ways to choose $(C_2, C_3)$ is $r_j \cdot (w_j - 1)$.

2. **Determining the maximum count $T$ of triples**:
- Utilize symmetry and combinatorial reasoning under constraints for maximizing white cells—for a balanced distribution, when $\frac{1}{3}$ of the cells are red helps achieving maximal overlap.
- Assume the table is partitioned such that $w = \frac{2}{3} \times 999^2$ white cells and $r = \frac{1}{3} \times 999^2$ red cells. This ratio balances the need for high overlap without inaccessible segments.

3. Calculating the number of such triples:
- Each red cell (given it belongs to both computations for row and column overlap) can contribute an additional count due to the distributed symmetry:
$$T \leq \left( \frac{1}{3} \cdot 999^2 \right) \left(\frac{2}{3} \cdot 999\right)\left(\frac{2}{3} \cdot 999\right).$$
- This expression represents the best overlapping distribution of white cells while minimizing wasteful triple counts, such as purely white rows or columns.

4. **Evaluating $T$**:
- Simplifying this we find:
$$T = \frac{4}{27} \cdot 999^4$$

Hence, by balancing the proportions of red and white cells and efficiently placing them within the grid, the maximum value of $T$ is given by:
$$\boxed{\frac{4}{27} \cdot 999^4}$$