In this paper the authors consider a pair of differential equations y 1 + p 1 ( x ) y 1 = 0 , y 2 + p 2 ( x ) y 2 = 0 {y_1} + {p_1}(x){y_1} = 0,{y_2} + {p_2}(x){y_2} = 0 , where p i ( x ) {p_i}(x) are positive and continuous, and where solutions y 1 ( x ) {y_1}(x) and y 2 ( x ) {y_2}(x) have common consecutive zeros at x = a x = a and x = b x = b . They show that if the curves y = p 1 ( x ) y = {p_1}(x) and y = p 2 ( x ) y = {p_2}(x) have a single intersection (possibly a closed subinterval) and if p 1 ( a ) > p 2 ( a ) , p 2 ( b ) > p 1 ( b ) {p_1}(a) > {p_2}(a),{p_2}(b) > {p_1}(b) , the first conjugate point of a + ε a + {\varepsilon } ( ε > 0 {\varepsilon } > 0 and small) for the second equation precedes that of the first.