Uranium-bearing minerals that give lead-uranium and lead—lead ages that are essentially in agreement, i.e. concordant, generally are considered to have had a relatively simple geologic history and to have been unaltered since their deposition. The concordant ages obtained on such materials are, therefore, assumed to approach closely the actual age of the minerals. Many uranium-bearing samples, particularly uranium ores, give the following discordant age sequences; Pb²⁰⁶U²³⁸<Pb²⁰⁷U²³⁵⪡Pb²⁰⁷Pb²⁰⁶ or, less frequently, Pb²⁰⁷Pb²⁰⁶⪡Pb²⁰⁷U²³⁵<Pb²⁰⁶U²³⁸. These discordant age sequences have been attributed most often to uncertainties in the common lead correction, selective loss of radio-active daughter products, loss or gain of lead or uranium, or contamination by an older generation of radiogenic lead.

The evaluation of discordant lead isotope age data may be separated into two operations. The first operation, with which this paper is concerned, is mechanical in nature and involves the calculation of the different possible concordant ages corresponding to the various processes assumed to have produced the discordant ages. The second operation is more difficult to define and requires, in part, some personal judgement. It includes a synthesis of the possible concordant age solutions with other independent geologic and isotopic evidence. The concordant age ultimately chosen as most acceptable should be consistent not only with the known events in the geologic history of the area, the age relations of the enclosing rocks, and the mineralogic and paragenetic evidence, but also with other independent age measurements and the isotopic data obtained on the lead in related or associated non-radioactive minerals.

The calculation of the possible concordant ages from discordant age data has been greatly simplified by Wetherill's graphical method of plotting the mole ratios of radiogenic Pb²⁰⁶U²³⁸ (N²⁰⁶N²³⁸) vs. radiogenic Pb²⁰⁷U²³⁵ (N²⁰⁷N²³⁵) after correcting for the contaminating common Pb²⁰⁶ and Pb²⁰⁷. The linear relationships noted in this graphical procedure have been extended to plots of the mole ratios of total Pb²⁰⁶U²³⁸ (^tN²⁰⁶N²³⁸) vs. total Pb²⁰⁷U²³⁵ (^tN²⁰⁷N²³⁵). This modification permits the calculation of concordant ages for unaltered samples using only the Pb²⁰⁷Pb²⁰⁶ ratio of the contaminating common lead.

If isotopic data are available for two samples of the same age, x and y, from the same or related deposits or outcrops, graphs of the normalized difference ratios[(N₂₀₆N₂₀₄)x − (N₂₀₆N₂₀₄)y(N₂₃₈N₂₀₄)x −(N₂₃₈N₂₀₄)y] vs. [(N₂₀₇N₂₀₄)x − (N₂₀₇N₂₀₄)y(N₂₃₅N₂₀₄)x −(N₂₃₅N₂₀₄)y] can give concordant ages corrected for unknown amounts of a common lead with an unknown Pb²⁰⁷/ Pb²⁰⁶ ratio. (If thorium is absent the difference ratios may be normalized with the more abundant index isotope, Pb²⁰⁸.) Similar plots of tho normalized, difference ratios for three genetically related samples (x − y) and(x − z), will give concordant ages corrected, in addition, for either one unknown period of past alteration or initial contamination by an older generation of radiogenic lead of unknown Pb²⁰⁷/Pb²⁰⁶ ratio.

Practical numerical solutions for many of tho concordant age calculations are not currently available. However, the algebraic equivalents of these new graphical methods give equations which may be programmed for computing machines. For geologically probable parameters the equations of higher order have two positive real roots that rapidly converge on the exact concordant ages corrected for original radiogenic lead and for loss or gain of lead or uranium. Modifications of these general age equations expanded only to the second degree have been derived for use with desk calculators.

These graphical and algebraic methods clearly suggest both the type and minimum number of samples necessary for adequate mathematical analysis of discordant lead isotope age data. This mathematical treatment also makes it clear that discordant lead isotope data alone cannot provide the basis for the choice of one of the possible concordant age solutions. The new equations, in particular, provide an incentive to improve our physical constants, analytical techniques and sampling methods in order that we may derive all of the useful geologic information that is available in a comprehensive lead isotope age study.