\[\begin{vmatrix}\operatorname{sn}z_{1}\operatorname{cn}z_{1}&\operatorname{cn}z%
_{1}\operatorname{dn}z_{1}&\operatorname{cn}z_{1}&\operatorname{dn}z_{1}\\
\operatorname{sn}z_{2}\operatorname{cn}z_{2}&\operatorname{cn}z_{2}%
\operatorname{dn}z_{2}&\operatorname{cn}z_{2}&\operatorname{dn}z_{2}\\
\operatorname{sn}z_{3}\operatorname{cn}z_{3}&\operatorname{cn}z_{3}%
\operatorname{dn}z_{3}&\operatorname{cn}z_{3}&\operatorname{dn}z_{3}\\
\operatorname{sn}z_{4}\operatorname{cn}z_{4}&\operatorname{cn}z_{4}%
\operatorname{dn}z_{4}&\operatorname{cn}z_{4}&\operatorname{dn}z_{4}\end{%
vmatrix}=0.\]