Suppose we have a radio-current of 100,000 cycles modulated by the frequencies in the voice range. We find in the output of our transmitting set not only a current of 100,000 cycles but currents in two other ranges of frequencies. One of these is above the signal frequency and extends from 100,200 to 102,000 cycles. The other is the same amount below and extends from 98,000 to 99,800 cycles. We say there is an upper and a lower "band of frequencies." All these currents are in the complex wave which comes from the radio-transmitter. For this statement you will have to take my word until you can handle the form of mathematics known as "trigonometry." When we receive at the distant station we receive not only currents of the signal frequency but also currents whose frequencies lie in these "side-bands." No matter what radio-frequency we may use we must transmit and receive side-bands of this range if we use the apparatus I have described in the past letters. You can see what that means. Suppose we transmit at a radio-frequency of 50,000 cycles and modulate that with speech. We shall really need all the range from 48,000 cycles to 52,000 cycles for one telephone message. On the other hand if we modulated a 500,000 cycle wave by speech the side-bands are from 498,000 to 499,800 and 500,200 to 502,000 cycles. If we transmit at 50,000 cycles, that is, at 6000 meters, we really need all the range between 5770 meters and 6250 meters, as you can see by the frequencies of the side-bands. At 100,000 cycles we need only the range of wave-lengths between 2940 m. and 3060 m. If the radio-frequency is 500,000 cycles we need a still smaller range of wave-lengths to transmit the necessary side-bands. Then the range is from 598 m. to 603 m. In the case of the transmission of speech by radio we are interested in having no interference from other signals which are within 2000 cycles of the frequency of our radio-current no matter what their wave-lengths may be. The part of the wave-length range which must be kept clear from interfering signals becomes smaller the higher the frequency which is being modulated. You can see that very few telephone messages can be sent in the long-wave-length part of the radio range and many more, although not very many after all, in the short wave-length part of the radio range. You can also see why it is desirable to keep amateurs in the short wave-length part of the range where more of them can transmit simultaneously without interfering with each other or with commercial radio stations.