The above condition for an ideal cascade may also be expressed in terms of abundance ratios:

їі’+i ~Vt-i — it

Figure 12.15 shows three stages of an ideal cascade in which this condition holds. From the
definition of the heads separation factor,

Vi = Ki

In an ideal cascade, because of (12.84),

Vi ~ РІі+i

Similarly, Vi*i=PVi

By multiplying these two equations together,

Vi+i = P2£i+i

But, from the definition of the separation factor,

Vi* і =

so that /3 = y/a = у

This relation between the heads and tails separation factors and the stage separation factor is the key property of an ideal cascade.

In the close-ftactionation case, in which (3—1 and a — 1 are small compared to unity,

An equation for the cut б in an ideal cascade is obtained from Eq. (12.12) by substituting for у and x their values in terms of z from Eqs. (12.18) and (12.20) and using the condition P = y: