The present paper, which can be considered as a continuation of the first author's previous papers [\textit{C. A. Stuart}, J. Math. Pures Appl., IX. Sér. 80, No. 3, 281--337 (2001; Zbl 1056.74019) and Proc. R. Soc. Edinb., Sect. A, Math. 132, No. 3, 729--764 (2002; Zbl 1020.34076)] is concerned with studying the buckling of a tapered rod. This physical phenomenon leads to the nonlinear eigenvalue problem \((A(s)u'(s))'+ \mu\sin u(s)= 0\) for all \(s\in (0,1)\), \(u(1)= \lim_{s\to 0}\, u'(s)= 0\) and \(\int^1_0 A(s) u'(s)\,ds 0\) for all \(s> 0\) and \(\lim_{s\to 0}\, A(s)/s^p= L\) for some constants \(p\geq 0\) and \(L\in (0,\infty)\). The authors deal with the critical case \(p=2\) and study the set of all solutions of the problem. In particular, they find the points \(\mu\in \mathbb{R}_+\) such that bifurcation ocurs at \((\mu, 0)\).