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Exceptional Integrals of a not completely Integrable Total Differential Equation

Published online by Cambridge University Press:  18 May 2009

H. T. H. Piaggio
H. T. H. Piaggio
Affiliation:
University of Nottingham
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1. There are exceptional integrals of the total differential equation

in the case when it is not completely integrable, and so when the invariant

is not identically zero, which do not seem to be mentioned by any standard authorities such as Cartan, Goursat, de la Vallée Poussin, and Schouten and Kulk. These are integrals of (1) which do not reduce I to zero. They arise only when the first partial derivates of P, Q, R are not all continuous. A simple example is z = 0 as an integral of

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 1 , Issue 3 , September 1953 , pp. 137 - 138
Copyright
Copyright © Glasgow Mathematical Journal Trust 1953

References

REFERENCES

(1)Cartan, E., Ann. Éc. Norm. Sup., (3), 16, 279, 280 (1899).Google Scholar
(2)Goursat, E., Leçons sur le problème de Pfqff (Paris, 1922), pp. 187–190.Google Scholar
(3)de la Vallée Poussin, Ch.-J., Cours d'Analyse Infinitésimale, t. II. (7th ed., 1946), pp. 303–4.Google Scholar
(4)Sohouten, J. A., and Kulk, W. V. D., Pfqff's Problem and its Generalizations (Oxford, 1949).Google Scholar