THE DIFFERENTIAL-EQUATION Q = 0 IN WHICH Q IS A QUADRATIC FORM IN Y'', Y', Y HAVING MEROMORPHIC COEFFICIENTS

A simple necessary and sufficient condition is given for the solutions of Q = 0 to be free of movable branch points. And, when the condition is satisfied, all the solutions of Q = 0 can be obtained by solving linear differential equations of order less-than-or-equal-to 2. There are four mutually exclusive cases. We shall relate Case 4 to less convenient conditions P. Appell had introduced. We shall also show how Cases 3 and 4 together motivated our discovery of an identity that is essential for a satisfactory theory of relative invariants for homogeneous linear differential equations.