It is shown that the disconjugate equation (1) L x ≡ ( 1 / β n ) ( d / d t ) ⋅ ( 1 / β n − 1 ) ⋯ ( d / d t ) ( 1 / β 1 ) ( d / d t ) ( x / β 0 ) = 0 Lx \equiv (1/{\beta _n})(d/dt) \cdot (1/{\beta _{n - 1}}) \cdots (d/dt)(1/{\beta _1})(d/dt)(x/{\beta _0}) = 0 , a > t > b > t > b , where β i > 0 {\beta _i} > 0 , and (2) β i ∈ C ( a , b ) {\beta _i} \in C(a,b) , can be written in essentially unique canonical forms so that ∫ b β i d t = ∞ ( ∫ a β i d t = ∞ ) {\smallint ^b}{\beta _i}dt = \infty ({\smallint _a}{\beta _i}dt = \infty ) for 1 ≤ i ≤ n − 1 1 \leq i \leq n - 1 . From this it follows easily that (1) has solutions x 1 , … , x n {x_1}, \ldots ,{x_n} which are positive in (a, b) near b ( a ) b(a) and satisfy lim t → b − x i ( t ) / x j ( t ) = 0 ( lim t → a + x i ( t ) / x j ( t ) = ∞ ) {\lim _{t \to b}} - {x_i}(t)/{x_j}(t) = 0({\lim _{t \to a}} + {x_i}(t)/{x_j}(t) = \infty ) for 1 ≤ i > j ≤ n 1 \leq i > j \leq n . Necessary and sufficient conditions are given for (1) to have solutions y 1 , … , y n {y_1}, \ldots ,{y_n} such that lim t → b − y i ( t ) / y j ( t ) = lim t → a + y j ( t ) / y i ( t ) = 0 {\lim _{t \to b}} - {y_i}(t)/{y_j}(t) = {\lim _{t \to a}} + {y_j}(t)/{y_i}(t) = 0 for 1 ≤ i > j ≤ n 1 \leq i > j \leq n . Using different methods, P. Hartman, A. Yu. Levin and D. Willett have investigated the existence of fundamental systems for (1) with these properties under assumptions which imply the stronger condition ( 2 ′ ) β i ∈ C ( n − i ) ( a , b ) (2’){\beta _i} \in {C^{(n - i)}}(a,b) .