This paper presents a method for determining the shapes of two overlapping functionsf1(x) andf2(x) from an observed set of additive mixtures, [αif1(x) + βif2(x);i= 1, …,n), of the two functions. This type of problem arises in the fields of spectrophotometry, chromatography, kinetic model building, and many others. The methods described by this paper are based on the use of principal component techniques, and produce two bands of functions, each of which contains one of the unknown, underlying functions. Under certain mild restrictions on thefj(x), each band reduces to a single curve, and thefi(x) are completely determined by the analysis.