Assume that a beam of light enters a material sample. Define z as an axis parallel to the direction of the beam. Divide the material sample into thin slices, perpendicular to the beam of light, with thickness d z sufficiently small that one particle in a slice cannot obscure another particle in the same slice when viewed along the z direction. The radiant flux of the light that emerges from a slice is reduced, compared to that of the light that entered, by d Φ Φ e ( z ) = − − μ μ ( z ) Φ Φ e ( z ) d z , where μ is the (Napierian) attenuation coefficient, which yields the following first-order linear, ordinary differential equation : d Φ Φ e ( z ) d z = − − μ μ ( z ) Φ Φ e ( z ) . The attenuation is caused by the photons that did not make it to the other side of the slice because of scattering or absorption. The solution to this differential equation is obtained by multiplying the integrating factor exp ⁡ ⁡ ( ∫ ∫ 0 z μ μ ( z ′ ) d z ′ ) throughout to obtain d Φ Φ e ( z ) d z exp ⁡ ⁡ ( ∫ ∫ 0 z μ μ ( z ′ ) d z ′ ) + μ μ ( z ) Φ Φ e ( z ) exp ⁡ ⁡ ( ∫ ∫ 0 z μ μ ( z ′ ) d z ′ ) = 0 , which simplifies due to the product rule (applied backwards) to d d z [ Φ Φ e ( z ) exp ⁡ ⁡ ( ∫ ∫ 0 z μ μ ( z ′ ) d z ′ ) ] = 0.