In the interesting new book by Carslaw and Jaeger, Operational Methods in Applied Mathematics, a point is raised which does not seem to have been explained adequately in this or any other book on this subject that I have seen. This concerns the application of initial conditions to switching problems in electric circuit theory. Reference is made to this deficiency in a review of this book in the Gazette, XXVI, No. 268 (February 1942), pp. 63-64, and the present article is a suggested way of overcoming it.

In this article the method of solution is applied to Bessel’s equation, but it can be used in the same way to solve other linear equations.

The note on this subject published in the Gazette of July 1941 was written in the hope that it might clear away a difficulty which I have good reason to know many students and teachers of dynamics find, when confronted by a problem of changing mass, where it is not obvious which, if indeed either, of the equations M dV/dt = F or d (MV)/dt = F applies. Among letters previously received on the subject I have found it asserted that the latter equation is “fundamental” and that results not in conformity with it must be wrong. I showed by three simple examples of changing mass that it is possible for one or other of these equations to be true or for neither to be true, and hoped that readers would feel satisfied to form the equation of motion for any such case directly, by the simple process of equating the change of momentum in time δt to the impulse of the force producing the change. Could there be a simpler procedure? Unfortunately readers in general are not so easily satisfied, and I seem to have stimulated an ambition to produce an equation of motion “of quite general validity” applicable to all such cases.