Abstract T he basic concepts of elastic stability are developed systematically for a structural system described by n generalized co-ordinates and a variable loading parameter. The expansion of the total potential energy function as a power series in a locally principal set of co-ordinates allows a critical equilibrium state and the neighbouring equilibrium paths to be studied analytically. The critical path configurations associated with the distinct phenomena of ‘snapping’ and ‘buckling’ are examined in detail. The snapping condition, in which an equilibrium path loses its stability on yielding a local extremum of the loading parameter, is the more general, and arises when a single stability coefficient vanishes. The buckling condition, in which a path loses its stability at a point of intersection (bifurcation), arises when a single stability coefficient and a second energy coefficient vanish simultaneously.