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that every relative integral invariant is the sum of an absolute integral invariant and the integral of a symbolic total differential, etc. The last three chapters are devoted to recent results in the study of systems of Pfaff, a great part of which is due to Cartan. The first question considered in Chapter 6 is as to the maximum number of dimensions p of an integral manifold of a system of r forms of Pfaff. There is a rich amount of material, too varied to lend itself to brief discussion, on systems of linear differential expressions, contact transformations, derived systems, the problem of Monge, second order partial differential equations, etc. The final chapter is devoted to the classification of the integral elements of a system of Pfaff and to the existence theorem. The book affords an excellent introduction to the study of a problem which occupies a very central position in the theory of differential equations. It gives a detailed survey of the classical theory, of its connections with other domains, of its most modern delelopments, and of the directions in which further advances may be made.

Book Review: A Treatise on the Theory of Bessel Functions