The relationship for $ T $ is given by:

$ T \propto S^\alpha A^\beta \rho^\gamma R^\delta $

To find constants $ \alpha $, $ \beta $, $ \gamma $, and $ \delta $, analyze the dimensional formulae:

Dimension analysis for Time $ T $:

$ \text{M}^0 \text{L}^0 \text{T}^1 = \left(\text{M} \text{T}^{-2}\right)^\alpha \left(\text{L}^2\right)^\beta \left(\text{M} \text{L}^{-3}\right)^\gamma \text{L}^\delta $

Equate dimensions:

$ \text{M}^0 \text{L}^0 \text{T}^1 = \text{M}^{\alpha + \gamma} \text{L}^{2\beta - 3\gamma + \delta} \text{T}^{-2\alpha} $

From comparing dimensions:

For mass (M): $\alpha + \gamma = 0$

For length (L): $2\beta - 3\gamma + \delta = 0$

For time (T): $ -2\alpha = 1 \Rightarrow \alpha = -\frac{1}{2} $

Solving equations:

Using $\alpha = -\frac{1}{2}$,

$ \alpha + \gamma = 0 \Rightarrow -\frac{1}{2} + \gamma = 0 \Rightarrow \gamma = \frac{1}{2} $

Substituting these into the equation for length:

$ 2\beta - 3\left(\frac{1}{2}\right) + \delta = 0 $

Assume $\beta = -1$ to solve for $\delta$:

$ 2(-1) - \frac{3}{2} + \delta = 0 \quad \Rightarrow \quad \delta = \frac{7}{2} $

By this analysis, the values are $\alpha = -\frac{1}{2}$, $\beta = -1$, $\gamma = \frac{1}{2}$, and $\delta = \frac{7}{2}$.