The primary purpose of this book is to develop the theory of systems of partial differential equations and that of pfaffian systems so as to exhibit clearly the relation between the two theories. The questions treated concern almost exclusively the existence of solutions and methods of approximating them rather than their properties, whose study seems to belong to the theory of functions.
In writing the book the author has been guided by a desire for generality in results and conciseness in subject matter and proofs. As a consequence, the postulational method seemed to force itself upon him. Roughly, the plan has been to take a few existence theorems as postulates and construct the theory upon them. A consistency proof is included by proving the postulates in particular cases. The original plan included extensions of the consistency proofs, but the pressure of other duties prevented carrying this out.
The ideas and nomenclature of modern algebra, as developed, for instance, in van dor Waerden's admirable treatise, have been freely used. Some modifications of certain topics, essential for our purposes, have been included, but no systematic development of the theory of commutative polynomial rings has been made. On the other hand, the theory of a certain non-commutative polynomial ring, called here a Grassmann ring, is developed in detail from the postulates in Chapter III, which together with Chapter IV develops ideas introduced by Grassmann and brought to such a high degree of perfection by Cartan. A combination of Cartan's notation, the tensor calculus, and modern algebraic concepts seems very effective. Incidentally, the results about determinants and linear dependence, which are needed, can be proved directly from the postulates as readily as the manner of stating them in the literature can be modified to fit the case in hand.
The treatment of the algebraic case is the author's. Although it has close connection through the highest common factor with Ritt's excellent discussion, which is based on the division algorithm, it differs radically in several respects from that work because of a difference in purpose and viewpoint. In the first place, the basis of our method is algebra, rather than analysis. Secondly, reduoibility, which plays such a prominent r61e in Ritt's developments, is of little importance in ours. With existence theorems as our chief objective, the important thing for us is to eliminate multiple roote. A polynomial's having two factors, for example, does not prevent the application of the implicit function theorem, if the factors are distinct, and making that theorem applicable is the chief purpose of the reduction process. Incidentally, it might be well to point out that the term "reducible" has slightly different meanings in the two theories. The system j/2, which Ritt classes as irreducible, is reducible in ours.
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