The authors deal with the transcendental equation of the type \[ (z+ pz+ q)\exp(\tau z)+ rz=0, \] where \(p\), \(q\), \(r\), \(\mathbb{R}\), \(\tau, p>0\), \(q>0\), \(r=0\), \(n=0,1,2\). The main results are concerned with the case \(n=0\), which is important in the stability theory of delay. A new necessary and sufficient condition for all the roots to lie to the left of the imaginary axis is given. The proofs are elementary and independent of Pontryagin's results. Applying the same approach in the cases \(n=1\) and \(n=2\), the authors give sufficient conditions for instability.