\frac{dq'}{dS} = -\sum_{i=1}^{n} \mathbf{J}_i(x_i, y_i, p', p'') \\ \frac{dx_i}{dS} = \frac{x_i \sum_{i=1}^{n} \mathbf{J}_i(x_i, y_i, p', p'') - \mathbf{J}_i(x_i, y_i, p', p'')}{q'} \\ y_i = \frac{\mathbf{J}_i(x_i, y_i, p', p'')}{\sum_{i=1}^{n} \mathbf{J}_i(x_i, y_i, p', p'')} \\ \text{At } S = 0, q' = Q^{in}, x_i = x_{in}