Partial melting of natural rocks is likely to proceed with the pooling of liquids which have been produced by a range of mass fractions of liquid development (O'Hara, 1985, 1995; Richter, 1986; Langmuir, 1989; Eggins, 1992; Plank & Langmuir, 1992; O'Hara & Fry, 1996). Partial crystallization of magmas in magma chambers may involve the integration of residual liquids from a range of mass fractions of liquid survival either through the crystallization zone or across the crystallizing surface (Langmuir, 1989; O'Hara & Fry, 1996). The concentrations of ideally behaved trace elements in an integrated liquid or solid and that in the product of a simple process operating at the same average mass fraction of liquid in the system are, in general, significantly different (O'Hara, 1995; O'Hara & Fry, 1996). However, under the highly selected and simplified conditions reviewed here, the results of integrating the liquids and solids produced in simple fractional melting or crystallization models are formally identical at all values of the distribution coefficient to the product of a simple equilibrium process at the average mass fraction of melt or residual liquid. The special conditions require that the mass fraction of the source solid or liquid which has been processed varies linearly with position within the zone of melting or crystallization; that the mass of source solid or liquid processed in each increment of position within the zone is a constant; that the limiting values of the mass fraction of liquid involved are zero and unity; and that the bulk crystal–liquid distribution coefficient, d, is assumed constant throughout each of the processes. Given the first two conditions listed above, the average mass fraction of liquid, fav, developed in linear planform circumstances must be (fmax + fmin)/2, and given the third condition this expression must have the value 0.5 (circular planform circumstances are mentioned briefly below). The expression for the relative concentration of an element in the corresponding residue or cumulate is then simply equation (1) multiplied by the value of the distribution coefficient. The basis for each of the derivations which follow is to integrate the concentration of the component multiplied by the mass of liquid or solid in which it occurs (i.e. find the amount of the element in the total liquid or solid) and divide this by the integrated amount of liquid or solid, yielding the required relative concentration in the integrated product. Only high-school mathematics is involved.