Twice the tone takes a quarter of the capacity
Every circuit has "a period of vibration of its own analogous to the period of vibration of a weighted spring" (568,178). On 3 July 1899 his secondary rang at "n = 20,700 per sec. approx." with 15 jars a side. The ground waves wanted 48 jars a side, more than the transformer could charge, and "An octave would require only 12 jars on each side ... the octave vibration had only 1/4 of the energy." Move the jars and watch his two laws do the arithmetic.
The magnifying factor is the tone over twice its decay
His rule: the rise is "directly proportionate to" inductance "and inversely to" resistance. On 2 January 1900 his extra coil had p = 585,000 and L = 0.018, so m = 585,000 × 0.018 ÷ 1 = 10,530, "falling to about 10" with the lamps' 1,000 ohms. Written as a complex tone ω = √(p² − γ²) − iγ with γ = R/2L, the same number is m = p/2γ: how many swings it rings before it fades.
Two circuits tuned alike, coupled tight, ring at two tones
He measured every part: secondary 9,023,000 cm and 1.382 ohm, primary 230,945 cm, mutual inductance 851,021 and 863,313 cm, secondary capacity 622 cm. Tuning the primary by his rule L₁C₁ = L₂C₂ takes 30 of his 0.0009 mfd bottles, and each circuit alone rings at 63,700 per second. Coupled at k = 0.59 they ring at about 50,500 and 100,000 per second, the two numbers he published in 1900.
Zero at one end, maximum at the other
A grounded coil about a quarter wave long: "at one terminal the potential would be zero and at the other maximum." Such a wire rings only at 1, 3, 5, 7 ... times its lowest tone. His own wire lengths and tones, put through speed = 4 × length × frequency, give the speed each wire would need, against his 186,000 miles per second.
One geometry: across at the speed of light, around at π/2 times it
His figures: a wave train of "0.08484 of a second" at "about four hundred and seventy-one thousand two hundred and forty kilometers per second" over the surface, the earth's diameter "an odd multiple of the quarter wave length", and "The lowest frequency would appear to be six per second." With π = 3.1416, c = 300,000 km/s and D = 12,726 km they are one picture.
A pattern pinned to the storm sweeps past a still detector
"It showed clearly the existence of stationary waves ... the point where the lightning struck the ground would be a nodal point." His detector restarted about every half hour as the storm drew off to about 300 km, and his patent gives the earth waves as 25 to 70 km long. On his nodal reading each restart is the storm moving on half a wave.
The rise is the wire it was threaded on
A ball alone loses capacity as it rises off the ground. His readings rose as he slid it up the vertical wire. Rebuilt from his coil, wire and ball, the circuit gives the rise: low down the coil top and the wire above crowd the ball; at the free end of the wire it takes the end's charge.
His own doubt: the fundamental or the first octave
For the iron-pipe structure he computed 464.8 cm from its parts. By resonance he read 339.7, 311.2, 310.6, 304.7, 305.0 and, with a new coil, 267.5 cm; with the rounded hood about 358 cm. On 26 November: "It now seems that the tuning ... was made to the first octave instead of to the fundamental tone. This is to be ascertained."
One more jet in one set than the other
His mercury break: jets of 9 and of 10 give "ninety makes and breaks" a turn. His rule, "one more nozzle in one set than in the other", makes every meeting a separate instant. Counts that share a factor meet fewer times: 8 and 10 give only 40.
One third or one half: which drag law
The turbine patent says the torque goes as the square of the fluid's relative speed, and that the work is greatest at half the fluid's speed. With work = torque × runner speed, the square law peaks at one third. Half is what his later law gives: drag "exactly proportionate to its velocity relative to the system" in a narrow gap.
The record
Fixed by his own numbers and his own laws, then the questions his records leave open. Nothing here is interpretation; where two readings fit his numbers, both stand.
Fixed
| What his numbers fix | From |
|---|---|
| The 3–4 July 1899 storm moved at 25 to 70 km per hour; the earth waves vibrated 4,300 to 12,000 times a second | Notes, 1904 article, 787,412; half-wave reading (full wave: 50 to 140 km per hour) |
| His earth figures are one geometry: diameter 12,726 km crossed at 300,000 km/s, surface speed π/2 times that, lowest tone 5.89 per second | 787,412; proved in Lean |
| His octave rule is exact: twice the frequency takes a quarter of the capacity and holds a quarter of the energy | Notes, 3 Jul 1899; proved in Lean |
| The ball's rise with height is the wire it was threaded on; within 2.4% at the top for the 30-inch ball | Notes, 23 Oct and 11 Nov 1899; the rebuilt circuit |
| His structure: computed 464.8 cm, read 267 to 358 cm, four times that if tuned to the octave; neither matches | Notes, Nov 1899 |
| His straight-wire capacities agree with his formula within a few per cent; the resonance method was sound | Notes, 28 Oct, 10 and 16 Nov 1899 |
| On 2 January 1900 his wire was 4% longer than the quarter wave: either the frequency was 4% lower or the wave ran 4% faster | Notes, 2 Jan 1900; both stand |
| His measured November coils ring at two tones an octave apart, about 50,500 and 100,000 per second | Notes, Nov 1899; his 1900 figures |
| His condensers, controllers, quarter-wave rule and magnifying rule were fixed before Colorado and carried into the patents after it | 1886–1928 patents |
Still open
| Open | Why |
|---|---|
| Which single wavelength belonged to 3–4 July | The patent gives the range from "a large number of observations" |
| Whether his structure tunings were to the fundamental or the octave | "This is to be ascertained"; nothing later settles it |
| The rise in the ball readings beyond the conductor model, largest for the 18-inch ball | Streamers and eddy currents in unslit foil; both untested |
| Ground or air at long range: the earth alone, or the earth with the upper atmosphere as a second plate | Tesla's pages and Marinčić (1978) side by side; untested |
| His 1916 interview in full | Only excerpts seen |
| The New York notes of 1900–1901 | Located (Nikola Tesla Museum, 2008), not read |