The asymptotic distribution of the eigenvalues of a degenerate elliptic operator CLAS NORDI~ 1. Introduction Let R be a Riemannian manifold of dimension n > 1 and class C2, let qo e C2(R) be real and such that 9 = 0 => grad ~ # 0 and such that ~o > 0 defines a compact part By of R. Let X gjkdxidx k be the metric of R and d V = g 89 (g = det (gik)) its volume element. Let L2(R~) be the real ttilbert space on /~ with norm square I u2d V. Let us interpret the degenerate differential operator J R~ as the Friedrichs extension associated with the two quadratic forms Rep "R~o and the real space CI(R~). According to Baouendi and Goulaouic [1], A = d~ is a non-negative selfadjoint operator on L2(R~) and (I q-A) -z is compact. Leg {~i}g be the eigenvalues of A associated with a complete set of eigenfunctions and let N(~) be the number of those eigenvalues which are ~ a. We are going to give an asymptotic formula for N(~) as k--> oo. Let dv be the volume element on S = aR~ with respect to the induced metric and let a/av be the unit interior derivative on S. Let con be the volume of the unit ball in R" and put r : (27t)l-nr f (aT/av) (1-n)/2dv. s (1) Finally, let