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I t is a familiar property of the expansion of a function in series of functions that the coefficients often may be expressed in terms of the coefficients of the Taylor series for the original function. This has been done explicitly for many specific cases with functions of a single variable. In this paper there is presented a method of obtaining more general results of this nature for functions of any number of variables defined by power series. The umbral calculus introduced by Blissard in his Theory of generic functions* has been used by Lucas and Bell among others as a convenient instrument in the manipulation of generating functions. The algebra of the umbrae has been discussed by Bell and some of the simplest properties of these will be used in the theory presented below. A function ƒ(x) defined by a power series x ƒ 0 ) = l^dn —

A formal expansion theory for functions of one or more variables