Main results: “If

${\lambda}_{1}$,

${\lambda}_{2}$ are zeros of

${J}_{\nu}^{\text{'}\text{'}\text{'}}\left(x\right)$,

$0\le \nu <1$,

${\lambda}_{1}<{\lambda}_{2}<{j}_{\nu 1}$, then

${\lambda}_{1}$ is a steadily decreasing function of

$\nu $ as

$\nu $ increases to 1 and

${\lambda}_{2}$ is a steadily increasing function of

$\nu $ as

$\nu $ increases to 1. Further, there exists a (unique) value of

$\nu ={\nu}_{0}$ such that

${J}_{\nu}^{\text{'}\text{'}\text{'}}\left(x\right)$ has two zeros when

${\nu}_{0}<\nu <1$ and none when

$0<\nu <{\nu}_{0}$. When

$\nu ={\nu}_{0}$,

${J}_{\nu}^{\text{'}\text{'}\text{'}}\left(x\right)$ has a douoble zero in

$0<x<{j}_{\nu 1}$;” “Let

${\lambda}_{1}\left(\nu \right)<{\lambda}_{2}\left(\nu \right)$ denote the zeros of

${J}_{\nu}^{\text{'}\text{'}\text{'}}\left(x\right)$ in

$0<x<{j}_{\nu 1}$,

${\nu}_{0}<\nu <1$. Then

${\lambda}_{1}\left(\nu \right)\downarrow 0$,

${\lambda}_{2}\left(\nu \right)\uparrow \sqrt{3}={j}_{11}^{\text{'}\text{'}\text{'}}$ as

$\nu \uparrow 1$” (Theorems 4.1 and 4.2 in the paper, respectively). Some inequalities for the zeros studied are also established. Here

${\nu}_{0}=0\xb7755578\cdots $;

${j}_{\nu 1}$ is, as usual, the first positive zero of the Bessel function

${J}_{\nu}\left(x\right)$ and

${j}_{\nu k}^{\text{'}\text{'}\text{'}}$ $(k=1,2,\cdots )$ stands for the positive zeros of the third derivative

${J}_{\nu}^{\text{'}\text{'}\text{'}}\left(x\right)$ and

${J}_{\nu}\left(x\right)$. The above results complete other ones found in another paper by the author and

*P. Szegö* [Methods Appl. Anal. 2, 103-111 (1995;

Zbl 0833.33003)].