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7. In Professor Moore's paper the following theorem is established : If in the interval 0 ^ x ^ 1 the function f(x) is continuous, save at a finite number of points at which it has a finite jump, and if in each interval of continuity it has a second derivative that is finite and integrable, then the development of f(x) in Bessel functions of order zero will converge uniformly to the value f(x) throughout the interval 0 ^=x^x0<. c, where c is the smallest value of x for which f(x) is discontinuous. I t is a well known fact that, under much wider conditions than those imposed here, the development will converge uniformly to f(x) throughout any closed interval that does not include nor reach up to a point of discontinuity of f(x), and does not reach up to the origin. The uniform convergence of the series in the neighborhood of the origin, however, does not seem to have been discussed.

Note on the groups generated by two operators whose squares are invariant