where H, e, and m are as defined above, and R is the radius at which the beam is extracted. From this equation we see that for a given type of ion (where e and m are constant), the energy depends on the diameter and strength of the magnet, but not directly upon the voltage applied to the dees. The number of revolutions that an ion can make in a conventional cyclotron is limited to about 70 to 100. This is due to a very curious effect: as an ion is accelerated, its mass increases! [This phenomenon is explained by Einstein's special theory of relativity (see Fig. 3).] Referring back to Eq. (1), we see that if the ion mass (m) does not remain constant, but rather increases, then the resonant frequency (f) decreases. But since the dee potential continues alternating at a constant frequency, an ion soon begins to arrive "late" at the dee gap. By the time it has made about 70 to 100 turns an ion is so badly out of phase that it is no longer accelerated. Suppose now that we want to obtain an energy of 10 Mev. Because an ion can make a maximum of about 100 turns, the accelerating potential would have to be about 100,000 volts. However, Professor Lawrence hoped to reach 100 Mev with the new 184-inch cyclotron. This meant that the accelerating voltage would have to be about 1,000,000 volts. Preventing such a high voltage from sparking promised to be one of many formidable engineering problems. [Illustration: Fig. 3. Graph showing how the mass of an object increases as its velocity approaches that of light.] FOOTNOTES: [1] The grants were as follows: Rockefeller Foundation--$1,150,000; John and Mary Markle Foundation--$25,000; The Research Corporation--$50,000. The University of California added a guarantee of $175,000 to bring the total building fund to $1,400,000. [2] In the first cyclotrons the electrodes were shaped like the letter "D." [3] We have the values H = 15,000 gauss, e = 4.8 × 10^{-10} electrostatic units, and m = 1.6 × 10^{-24} gram. To find f, we write 15,000 (4.8) 10^{-10} f = ---------------------------------- , 2 (3.14)(1.6) 10^{-24} (3) 10^{10} f = 23.7 Mc. THE PRINCIPLE OF PHASE STABILITY Fortunately, Drs. Veksler and McMillan showed that relatively low dee voltages can be used to accelerate ions to very high energies. This is possible if the oscillator frequency is continuously decreased to keep it in synchronism with the decreasing rotational frequency of the ions. This would allow an ion to make many revolutions without becoming out of phase. This principle of phase stability was experimentally verified with the 37-inch cyclotron before being incorporated into the design of the 184-inch machine. Because it utilizes this principle, this machine has usually been referred to as a "synchrocyclotron" or "frequency-modulated cyclotron." However, it is sometimes called simply a "cyclotron."