expressed in terms of characteristic roots. The index is the number of negative roots λ which together with some set (z) (0) satisfy the system

and the nullity in the number of zero roots λ. In either case roots are counted according to their multiplicities, that is, the dimension of the associated set (z).

With this in mind suppose first that all critical points off are non-degenerate that is, of nullity zero. Then they are finite in number. Suppose that M_i is the number of index i. Suppose that R_i is the rank of the ith homology group H_i(M). Then Morse established the inequalities that bear his name, to wit

The second inequality is essentially the Birkhoff mini-max principle,³ and Morse in (1925) credits it with being a source of his idea for the inequalities. Poincaré was aware of the equality when n = 2.

The Morse inequalities yield the weaker but very useful inequalities

Morse's initial treatment in (1925) was of an even simpler case than the one just described. The space M was a bounded region in euclidean n-space and the boundary