The paper first digests existing information about 3rd-harmonic fields, mainly published previously in piecemeal fashion, and then adapts and extends it into modern form for use in symmetrical-component theory as applied to machines. This theory has provided the basis for a very successful 3:1 pole-changing machine which is described in another paper,8and the present paper includes a number of test results which verify that the theory of 3rd-harmonic fields given here is essentially sound.