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The synthesis of a current sawtooth from a limited number of first harmonics has been studied. It was found that good linearity is easier to obtain than a fast retrace. Scan distortion may be held below 5 per cent over 80 per cent of scan, by adding only 4 harmonics in a "least square" proportion. Fast flyback, on the other hand, requires 8 harmonics to be usable for commercial television. Several practical systems for resonant sweep have been tested. All circuits used shock excitation of multiresonant networks by pulses of current. Both pentodes and small hydrogen thyratrons were used successfully. The best multiresonator for synthetic sweep is a delay-line, 3 per cent shorter than a halfwave long and shorted at the far end. To minimize dispersion, a "slanting wafer" type of construction has been developed which permits control of mutual without effect on self-inductance.