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This paper is an extension of an article by V. Y. Savelyev appearing under the title "On the Theory of the Klystron," in the Journal of Technical Physics (Russian), vol. 10, no. 16; 1940. Appreciation is here expressed for the permission granted to The Institute of Radio Engineers to present portions, modifications or extensions of this material in the pages of its PROCEEDINGS. Savelyev's work is a review of some of the results of Webster, and seems to present a simpler method of development, especially in respect to electron-repulsion effects in the beam. In this article the use of the kiystron as an amplifier and as an oscillator, and the optimum operating conditions are considered. The restricting assumption is that the depth of modulation be small, and the time of flight of an electron through the buncher is neglected. It is found that, in the case of an amplifier, the presence of electron-repulsion debunching sets a limit to the upper value of length of drift space which can be used. Thus the gain is limited. In the case of the oscillator operating under optimum conditions the debunching has very little effect.