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In 1961, J. Barrett showed that if the first conjugate point 
${{\eta }_{1}}\left( a \right)$ exists for the differential equation 
${{\left( r\left( x \right){y}'' \right)}^{\prime \prime }}=p\left( x \right)y$, where 
$r\left( x \right)\,>\,0$ and 
$p\left( x \right)\,>\,0$, then so does the first systems-conjugate point 
${{\hat{\eta }}_{1}}\left( a \right)$. The aim of this note is to extend this result to the general equation with middle term 
${{\left( q\left( x \right){y}' \right)}^{\prime }}$ without further restriction on 
$q\left( x \right)$, other than continuity.
