The use of a matrix Riccati equation to establish sufficiency theorems in the calculus of variations is well known (see [3], e.g.). In this note we extend the method to give an elementary proof of the Morse index theorem, thereby eliminating the ad hoc subdivisions of the classical proof (see [2], [4], or [5], e.g.). Although we consider only the simplest form of the index theorem, the technique can easily be adapted to generalizations such as that of [N]. Let E be a euclidean space, G the linear space of broken CIO maps u: [0, T]-->E for some closed interval [0, T], and H the subspace of those uEG such that u(0) =u(T) =0. Let P be a given Coo map of [0, T] into the selfadjoint linear transformations of E, and let U: [0, T]->HomR(E, E) be the unique Coo map satisfying U(O) =0, U' (0) = I, and +P U= 0; U can also be regarded as a linear transformation of G into itself which is stable on the subspace H. The multiplicity of any t& (O, T] is the nullity of U(t), and t is a focal point whenever it has positive multiplicity. The index form I on HXH is defined by I(u, v)-=ft 'o[(u', v')-(Pu, v)]dt, and an inner product J on HXH is defined by J(u, v) =fT0(u, v)dt, where (, ) is in both cases the inner product on E XE; for convenience we set I(u, u) = I(u) and J(u, u) =J(u). The index i(I) is the dimension of any maximal subspace of H on which I is negative definite, and the nullity n(I) is the dimension of the subspace of those u CH such that I(u, v)=0 for all vEH. The first portion of our proof of the index theorem is standard, depending on the following classical lemma: