$$A$$, $$B$$, $$C$$, and $$D$$ are four different physical quantities having different dimensions. None of them is dimensionless. But we know that the equation $$AD = C\ln(BD)$$ holds true. Then which of the combination is not a meaningful quantity?

Given that $$A$$, $$B$$, $$C$$, and $$D$$ are four different physical quantities with different dimensions and none is dimensionless, and the equation $$AD = C \ln(BD)$$ holds true. For this equation to be dimensionally consistent, the argument of the logarithm must be dimensionless. Therefore, $$BD$$ must be dimensionless, which implies $$[B][D] = 1$$, so $$[B] = [D]^{-1}$$.

The logarithm $$\ln(BD)$$ is also dimensionless since its argument is dimensionless. Thus, the right-hand side $$C \ln(BD)$$ has dimensions $$[C]$$. The left-hand side $$AD$$ must have the same dimensions, so $$[A][D] = [C]$$.

Now, we need to determine which of the given options is not a meaningful quantity. A meaningful quantity requires all terms in an expression to have the same dimensions, allowing operations like addition and subtraction.

Option A: $$\frac{C}{BD} - \frac{A^2 D^2}{C}$$

Since $$BD$$ is dimensionless, $$\frac{C}{BD}$$ has dimensions $$[C]$$. For $$\frac{A^2 D^2}{C}$$, substitute $$[C] = [A][D]$$: dimensions are $$\frac{[A]^2 [D]^2}{[A][D]} = [A][D] = [C]$$. Both terms have dimensions $$[C]$$, so the expression is meaningful.

Option B: $$A^2 - B^2 C^2$$

$$A^2$$ has dimensions $$[A]^2$$. For $$B^2 C^2$$, substitute $$[B] = [D]^{-1}$$ and $$[C] = [A][D]$$: dimensions are $$[D]^{-2} \cdot ([A][D])^2 = [D]^{-2} \cdot [A]^2 [D]^2 = [A]^2$$. Both terms have dimensions $$[A]^2$$, so the expression is meaningful.

Option C: $$\frac{A}{B} - C$$

$$\frac{A}{B}$$ has dimensions $$\frac{[A]}{[B]}$$. Substitute $$[B] = [D]^{-1}$$ and $$[C] = [A][D]$$: $$\frac{[A]}{[D]^{-1}} = [A][D] = [C]$$. Both terms have dimensions $$[C]$$, so the expression is meaningful.

Option D: $$\frac{A^2 - AC}{D}$$

First, examine the numerator $$A^2 - AC$$. $$A^2$$ has dimensions $$[A]^2$$. $$AC$$ has dimensions $$[A][C] = [A] \cdot [A][D] = [A]^2 [D]$$ (since $$[C] = [A][D]$$). Since $$[D]$$ is not dimensionless (as $$D$$ is not dimensionless and all quantities have different dimensions), $$[A]^2 \neq [A]^2 [D]$$. The terms $$A^2$$ and $$AC$$ have different dimensions, so their subtraction is not allowed dimensionally. Thus, the expression is not meaningful.

Hence, the correct answer is Option D.