1. A mathematical theory of the culling process in dairy cattle has been developed. This was necessary to give a conceptual framework for the discussion of longevity in the dairy herd as a heritable character, of its genetic relationship with milk yield and of the effects of selection on the genetical analysis of yield itself.

2. Culling is assumed to take place at the end of each lactation by truncation selection of a normally distributed variate (the culling variate) of which yield itself is a component.

3. Expressions are then derived for the effect of the culling on various parameters of yield and for survival as a function of yield.

4. The derivation of expressions relevant to progeny groups is dependent on the relative values of the genetic and phenotypic regressions of the culling variate on yield.

5. Because little is known of the actual causes of culling, it is not permissible to predict the genetic gain in yield by multiplying the selection differential by the heritability. This is only justified if the genetic and phenotypic regressions of the culling variate on yield are equal.

6. A direct estimate of the genetic gain by selection can, however, be obtained from the covariance for any character between the mean relative survival of progeny groups and their mean value for the character.

7. The effect of culling on the genetic analysis of yield in later lactations is discussed, as is the effect of later culling on survival to later ages as a function of early yield. It is suggested that the regression coefficient of relative survival of progeny groups on heifer yield will be approximately linear with age whereas that of individuals will probably increase to a limiting value.

8. The heritability of survival to different ages will probably pass through a maximum at the fourth lactation.

9. Longevity itself, as a function of heifer yield, is the sum of the cumulative survival curves to each individual lactation. The regression of length of life on heifer yield, at the individual or at the progeny test level, can be related to the mean phenotypic and genetic selection applied to animals in the herd at any time.

10. Because culling is to a large extent a voluntary action of the farmer, longevity in the dairy herd, though it is under continual selection and may have a significant heritability, need not necessarily increase.