It is known [Herman Weyl, 1910] that every linear second-order differential expression L (with real coefficients) is such that Ly = λy(im λ ≠ 0) has at least one solution belonging to the class [Formula Ommitted] of functions, the squares of whose moduli are Lebesgue-integrable on [0, oo). This celebrated result was later proved by E. C. Titchmarsh (1940–1944), using sophisticated analysis of bilinear transformations. The aim of the present note is to prove the same result once again, but using only elementary analysis and school geometry. The power of this method will be appreciated further when one realises the amount of simplifications that can be achieved by this method in case of higher order expressions. This part of the note of course will be taken up in a subsequent paper. 1980 Mathematics subject classification (Amer. Math. Soc.) (1985 Revision): 34 B 20. © 1989, Australian Mathematical Society. All rights reserved.