Monday, December 21, 2015

Wrapping Up: Results from the Stanford Plasma Physics Lab

Hi all,

This is post is meant to summarize the work I've done at the Stanford Plasma Physics Lab on my water explosion project. School and other pursuits have occupied the majority of my time since my last post, so I've only been working on the project on-and-off since then. However, I was able to finish the experiments I set out to do over the summer of 2014, and I intend to report the results here.

The water does seem to contribute internal energy to the explosions. More and different research is needed to shed light on exactly how this happens.

To pick up where I left off last post:
  • I won the $1500 research grant I applied for, and used the money to buy a dedicated capacitor discharge circuit (here, from Information Unlimited), independent of the capacitor bank for the lab's plasma gun. The capacitor bank is 100 uF and can withstand up to 5 kV, though I've only ever charged it to 3 kV
  • I decided to wait on the shock-wave-time-of-flight-based measurements for explosion energy since:
    • the calorimeter-based measurements yielded satisfactory results, and
    • I don't currently know much about solving partial differential equations numerically.
Using my custom-made, hand-milled acrylic calorimeter, my custom thermistor feedthrough, Information Unlimited's capacitor discharging setup, and several high-current and high-voltage probes, I was able to get several complete measurements of a water explosion's input and output energies. And I have videos to prove it:

Notice that there are visible sparks around the chamber in both cases, and in the second video there's debris in the water following the explosion (compare the frames at 0:18 and 0:20). My guess is that this is the steel electrodes (or something on them) degrading. I tried estimating the expected energy released from ablating steel in water, to see if this might be a significant contribution to the energy recorded by the calorimeter.

Based on measurements from this article and currents typically seen in the discharge loop, I gathered that about 0.1 milligrams of steel is eroded from the electrodes in each shot. (This might not seem to be enough to color the water as it does, but note that the steel that does get eroded gets eroded very thoroughly). The standard enthalpy of formation of iron(iii) oxide is -824 kJ/mol and the molar mass of most steel is around 56 g/mol. Thus, we should expect to release on the order of 824/56*0.1 = 1.5 J from oxidizing the steel in the electrodes, which is negligible given our input energies.

Typical current and voltage waveforms for a shot through water
Calorimeter temperature over time
I was able to get data from five different shots before my acrylic explosion chamber gave out on me (read: exploded). In each shot, I charged the 100 uF capacitor bank to approximately 3 kV, for a total stored energy of ~450 J.

Table 1: Input and output energies for the measured water explosions.
(You can read my previous post for more details about my methods on calculating these energies).

As you can see, the net explosion energy exceeded the input pulse energy in every shot observed. In shot 1, the measured explosion energy even exceeded the total energy stored by over 200%.

Some points about table 1:
  • Both input pulse energies and output explosion energies are highly variable, and do not seem to be directly correlated.
    • Shots 1 through 3 have similar pulse energies but widely different explosion energies.
    • Shot 4 has the lowest pulse energy but the second highest explosion energy.
  • Shots 1 through 3 and 4 through 5 seem to be in two different classes as far as input pulse energy is concerned. This may have to do with the amount of contact resistance or dissipation in the circuit, which may change from shot to shot.
  • In each shot I charged the 100 uF capacitor bank to 3 kV, for a total stored energy of 450 J. Thus the efficiency of energy transmission from capacitor bank to calorimeter seems to vary significantly from about 80% in shot 1 to about 6% in shot 4.
  • The uncertainties on each measurement were calculated by propagating errors. I calibrated the calorimeter by submerging it in an ice bath, heating it with a known wattage, and measuring the resulting temperature change. From this, I estimated the uncertainty in the calorimeter's temperature measurements. I used this and the tolerance ratings on my voltage and current probes to compute the uncertainties displayed in table 1. 
My conclusion from these results: there's definitely something going on here. Unfortunately, I don't have the time, resources, or chemistry background to figure out what. I can, however, leave you with an unsubstantiated theory. I've speculated before (at the end of the post) on what mechanism might underlie these water explosions, but I've since come across some new papers and have some new ideas.

Graneau thought the formation of droplets in the water explosion was crucial to the release of energy in a water explosion. He attributed a the energy release to a difference in the heat capacities of bulk water and steam. However, heat capacity is a macroscopic thermodynamic property resultant from the statistical behavior of a large number of molecules in a bulk phase. Such macroscopic properties lose meaning at a microscopic scale, and if the explosions' strange properties are indeed due to the formation of micron-scale water droplets, then a microscopic explanation is probably required.

In the past, I proposed that the rapid formation of water droplets allows "weak" hydrogen bonds to collapse to "strong" hydrogen bonds in the water. Sun et al. (2013) found evidence that the lengths of both inter- and intra-molecular bonds in water depend on the number of neighbors the water molecule has. Water molecules with fewer than 4 neighbors tend to have longer hydrogen bonds and shorter covalent bonds. This has the net effect of making the bond "stronger," i.e. the bond has a more negative binding energy. Thus, when you have a lot of water molecules with a lower-than-average number of neighbors, as in a droplet or near an interface, you will see some rather unusual bond reorganization.  Water explosions are not the first bizarre phenomenon this bond lengthening and contracting might help explain; it's also been implicated in the Mpebma effect.

Illustration of two possible hydrogen-covalent bond lengths: the "weak" red state and the "strong" blue state
Figure from Zhang et al. (2013)

There is further evidence that water behaves weirdly at interfaces. Gerard Pollack of the University of Washington (check him out here) has done extensive research on water chemistry and has proposed the existence of a "fourth phase" of water, something between the liquid and solid phase, that forms at interfaces between bulk water and some hydrophilic surface. He describes the phase as a gel-like layer with an unusually high pH. It forms in layers of a few hundred microns, which Pollack calls "exclusion zones" since they tend to exclude solutes. These exclusion-zone layers likely form on the droplets created by a water explosion and may play an important role in the explosions' properties.

Gel-like “exclusion zone” boundary layer, from Pollack,  (2013)

If a water explosion releases energy, the second law of thermodynamics requires that the water produced by the explosion be something other than normal water; burning gasoline doesn't produce more gasoline. Pollack's "fourth phase" seems like a good candidate for the byproduct of a water explosion.

I've proposed two theories for the mechanism behind water explosions, but both of these theories still need substantial experimental and theoretical expansion. For example, Zhang et al. (2013) predict that there is a temperature dependence on the bond contraction; the O-H covalent bond contracts more (i.e. binding energy becomes more negative) when the water is at a higher temperature. Furthermore, Pollack has shown that incident light tends to expand the exclusion zone. Both of these observations suggest further experiments (e.g. firing the explosions at different water temperatures and at different exposures to light) that may shed more light on the nature of these explosions. However, as I've said, I don't currently have the time to pursue these or other experiments further. 

If anyone out there is interested in exploring this project further, please comment below and we can chat. Otherwise, I imagine this will be the last update I make to this blog for a while. Thanks for reading! This project has been a lot of fun. 

[1] Comment on “A Theory of Macromolecular Chemotaxis” and “Phenomena Associated with Gel–Water Interfaces. Analyses and Alternatives to the Long-Range Ordered Water Hypothesis” G. H. Pollack, The Journal of Physical Chemistry B 2013 117 (25), 7843-7846 DOI: 10.1021/jp312686x 

[2] C.Q. Sun, X. Zhang, X. Fu, W. Zheng, J.-l. Kuo, Y. Zhou, Z. Shen, and J. Zhou, Density and phonon-stiffness anomalies of water and ice in the full temperature range. J Phys Chem Lett, 2013. 4: 3238-3244. 

[3] Zhang, X., Huang, Y., Ma, Z., & Sun, C. (2013). O:H-O Bond Anomalous Relaxation Resolving Mpemba Paradox. arXiv:1310.6514.

Wednesday, October 8, 2014

New Set-Up at Stanford Plasma Physics Lab

I started at Stanford as a freshman last fall, and amongst the tumult of beginning college, I was able to navigate my way into an internship at the Stanford Plasma Physics Lab (the SPPL, check them out here). I spent 10 weeks this past summer working in the lab, and one of the several things I did there was build a new experimental apparatus for continuing my high school water explosion project. My work there is far from finished, and I have yet to take data from the new apparatus I set up, but I thought it was about time to let you guys know what I've been up to!

My favorite part about working at the SPPL has been getting to talk with other bright, interested scientists and engineers about water explosions and the eyebrow-raising papers that have been published on them. In particular, my conversations with Professor Mark Cappelli, the head researcher for the SPPL, have been very helpful in shaping my experimental design. Cappelli is an expert in applied plasma physics, and has worked extensively with Hall-effect thrusters, surface plasmons, and coaxial plasma accelerators, among other technologies. He has offered much guidance in crafting my experiment design since I first approached him about studying water arc explosions in his lab last spring.

From my conversations with him about the nature of the water explosions and the resources available to the lab, I decided to pursue a different method for measuring the water explosions' energies: bomb calorimetry. Instead of the projectile-based methods for measuring explosion energy that Graneau et al. used in their experiments (Graneau et al., 2000), I opted to make more precise energy measurements by setting off the explosions in an insulated, closed calorimeter and measuring how much energy the explosions deposit to the water as heat.

To produce the explosions, I used the capacitor banks and discharge circuit Professor Cappelli had set up for his high-current coaxial plasma gun. I worked with Keith Loebner and Ben Wang, two of Cappelli’s PhD students, to modify the system’s charging power supply, triggered spark gap switches, and safety interlocks to be compatible with my calorimeter.

One of the SPPL's four 52 uF, 20 kV capacitor banks. So far, I've just been using one of the four 14 uF caps shown here
My experiment is simple in principle: measure the energy going in to the chamber as electricity, measure the energy released from the explosion, and compare the two. I will measure the electrical energy delivered to the water as I did before: by monitoring the voltage across and the current through the chamber over the duration of the pulse event.

Measuring the energy released by the explosion will be more difficult. I currently have two different strategies for making this measurement.

Method 1: Closed Calorimeter
The first, and simplest, approach is to monitor the temperature of the water in the calorimeter over the duration of the explosion event. If the calorimeter is sufficiently well insulated, eventually all the various forms of energy (kinetic, thermal, chemical, etc.) released by the explosion will appear in the water as heat, and the water’s temperature will increase correspondingly over time. With this temperature-vs-time measurement, some heat loss modeling and a careful calibration of the chamber-water-system’s heat capacity, I can calculate the explosion’s energy straightforwardly.

The water explosion calorimeter, as of August 2014.

View of the inner chamber, with electrodes visible

I designed and machined the calorimeter myself in the lab’s machine shop. The brass piece mounted through the top permits thermistor and thermocouple measurements of the water’s temperature while still allowing the chamber to hold pressure.

Method 2: shock wave time-of-flight measurements
I may also be able to calculate the energy released in the water explosion indirectly by measuring the speed of the shockwave the explosion produces. In the late 1940s, Geoffrey Taylor derived an elegant expression for the energy released in an intense explosion involving only the shock front’s speed, the surrounding medium’s density, and the surrounding medium’s specific heat ratio (Taylor, 1950; Sedov, 1959). Taylor used his equations to calculate the energy released in the first atomic explosion in New Mexico based on images of the shock wave it created. I'm hoping to use Taylor's equations similarly, but on a water explosion instead of an atomic bomb.

Unfortunately, Taylor’s method relies on a self-similar solution to the fluid equations that is valid only for an ideal gas, and not for a medium like water without a polytropic equation of state. Last summer I attempted to reconcile Taylor’s methods with a couple different equations of state for water. I was able to make some progress, but it is unclear if my approach will pan out in the end.

To measure the shock wave’s speed in water, I will use either a high-speed Schlieren camera or an array of lasers and photodetectors to image the density gradient produced by the shock.

It is also possible that the shock wave will propagate out through the water and calorimeter and into the surrounding air. I could then use an array of microphones or other pressure transducers to measure the shock wave’s speed in air and apply Taylor’s methods directly. The hurdle with this approach is understanding the shock front’s behavior as it transitions between different media.

I will attempt to implement Method 2 only if Method 1 does not satisfactorily yield the desired energy measurements.

I've applied for one of Stanford's $1500 undergraduate research grants to support my continued work on my water explosion project throughout the school year. At the time of my writing this, I have yet to hear back from on high whether or not my application has been accepted. If I get the money, I'll be able to start working in the lab again within the end of the year. If not, I might have to wait until next summer to start up work on the project again. Either way, you'll be hearing back from me before too long!

Before I sign off, I'll leave you with a few other pictures of things around the lab this summer.

My initial prototype of the calorimeter. As you can see, I didn't make the walls quite thick enough. These water explosions are forces to be reckoned with!

My prototype of the thermistor amplifier circuit for measuring the temperature of the water in the calorimeter.
My ad-hoc power supply control unit and discharge triggering circuit

As an end note, I'd like to acknowledge a number of people at the SPPL whose help was invaluable this summer. Thanks to Keith Loebner and Ben Wang for their guidance and supervision, Mike Carter for his tutelage in the machine shop, and Mark Cappelli for his thoughtful feedback and discussions.

Graneau, P., Graneau, N., Hathaway, G., & Hull, R. (2000). Arc-liberated chemical energy exceeds electrical input energy. Journal of Plasma Physics63, 115-128.

Sedov, L. (1959). Similarity and Dimensional Methods in Mechanics. Academic Press. New York, NY.
Taylor, G. (1950). The formation of a blast wave by a very intense explosion: I. Theoretical discussion. Proceedings of the Royal Society of London, 159-174.

Tuesday, May 28, 2013

The Data are In!

Note: In this post I'll be referencing a couple papers, citations for which can be found at the post's end.

It's been a long time and a couple of science fairs since my last post, so I thought I'd take the time to update you on what I've been up to these past few months. With the end of the school year (and my high school career!) consuming a lot of my time, I haven't gotten to work on the project as much as I would have liked; in fact, I'm still not done experimenting or collecting data. I'll summarize the data I've collected so far and the methods I've used to collect it below.

A sample water arc explosion. Note the large fog jet and the bright arc inside the accelerator barrel.
Since I got my water explosion machine working reliably, I've been primarily concerned with quantifying the water explosions' energy inputs and outputs. The former has been relatively straightforward, as it only requires a couple of oscilloscope measurements (below). In each explosion I looked at, I measured the capacitor voltage before and after discharge, which I used to calculate the electrical energy delivered to the explosion chamber, as well as the peak discharge current, which is predicted to correlate with explosion strength (Graneau et al., 2000).

Sample capacitor voltage waveform. Each horizontal division represents 5 μs and each vertical division represents 5 kV. The voltage before discharge is about 15 kV.

Sample discharge current waveform. Each horizontal division represents 5 μs and each vertical division represents approximately 2.5 kA. The peak current is 4.5 kA.

I've made some minor tweaks to my charging circuit. I've also made this handy illustration to show them off (click for higher resolution):

The the EMF-sensitive shut-off circuit, which cuts power to the H-bridge inverter when it detects that the capacitors have discharged, now triggers more reliably and the switching transistors (in the H-bridge) are less prone to overheating. 

The latest incarnation of my water explosion chamber is pictured below:

The chamber comprises a cylindrical cup with an open top and two electrodes. One electrode projects through the bottom of the cup and the other forms a ring around the inside of the chamber. It's made from a giant block of wood because all the plastics I had available weren't strong enough to make a durable explosion chamber out of.

To determine a water explosion’s kinetic energy, I employed the same projectile-based method used by physicist Peter Graneau in his research on water explosions (Graneau et al., 2000). The idea is to place a porous projectile (mine was made of balsa wood) over the muzzle of the accelerator barrel so that a water explosion within the barrel launches the projectile into the air and some of the fog penetrates into the projectile during the process. The height to which the projectile ascends can be measured and used to estimate the initial kinetic energy (in the vertical direction) of the explosion.

Now for the maths: Let the balsa wood projectile have mass M and let the explosion launch it to a height h. Let m0 be the total mass of the expelled water, m1 be the mass of the water that bounces off the projectile, and m2 be the mass of the water absorbed by the projectile. The collision of m1 with M is elastic, and the collision of m2 with M is inelastic. Say m1 has initial average velocity μ01 and m2 has initial average velocity μ02. After the collision of m1 with M, say m1 has average velocity μf1 and M has velocity v1. After the collision of m2 with M, say the combined mass M+m2 has velocity v2.

M, m0, and m2 can be determined by before and after measurements of the projectile mass and water mass. h can be measured by taking a video recording of the explosion and analyzing the video frame-by-frame. The other quantities of interest, namely m1, μ01, and μ02, must be found indirectly.

Energy and momentum conservation in the collision of m1 with M require that:

Eqn. 1

and momentum conservation in the collision of m2 with M requires that: 

Eqn. 2
Examining the projectile, it follows from the conservation of mechanical energy that: 

Eqn. 3
where g is the acceleration due to gravity. Solving (3) for v2 yields:

Eqn. 4
By making some assumptions about the fog jet’s mass and velocity distributions, we can use (1), (2), and (4) to solve for m1, μ01, and μ02 and then calculate the fog explosion’s kinetic energy.

We can obtain a variety of approximations for the explosion’s kinetic energy using different models for the explosion’s velocity distribution. Each of the below velocity-mass functions could plausibly describe a water arc explosion given the behavior of the fog jets in the high-speed images presented in Hathaway et al. (1998).

Sample graphs of the velocity-mass function v(m) for a water explosion. The first has decay rate k = 5, while the second has k = 50
Eqn. 5

Using the methods described above, I was able to compute the input and output energies of several water explosions. A sampling of the data I collected is presented in Table 1 below (click to enlarge).

Table 1: The input and output energy data of 8 shots using distilled water at room temperature. In all shots, the charging capacitance C was 0.6 μF. The capacitor input energy was computed as E0 =  C/2*(V0^2 - Vr^2).
Key: V0: initial capacitor charging voltage; Vr: residual capacitor voltage after discharge; E0: electrical input energy; Ef: kinetic output energy; Ef/E0: the explosion’s efficiency, the ratio of output to input energy. M, m0, m2, and h are as defined in the Methods section. All kinetic energies are computed using a v(m) model with k = 300.

The results in Table 1 indicate that water arc explosions liberate stored potential energy (though these results are only as good as the assumptions made in obtaining them).  Looking at the last 3 columns of Table 1, it can be seen that the water explosion’s kinetic energy was even greater than the pulse discharge’s electrical energy in three of the eight shots displayed. These results are consistent with the findings reported in Graneau et al. (2000) and require that some sort of internal water energy is released during a water arc explosion.

If electrodynamic forces alone were responsible for the explosions, their kinetic energies should be only a fraction of the total input energy. For a typical shot with E= 60 J, a peak discharge current of 10 kA, and a pulse width of 10 μs,  the upward Lorentz pinch thrust should be about 0.1 N on average, corresponding to an explosion kinetic energy of 2.5 nJ = 2.5 x 10^(-9) J. This is about ten orders of magnitude off the observed values for Ein Table 1.

Note that the accuracy of a kinetic energy measurement depends largely on the explosion’s actual velocity-mass distribution. If this actual distribution differs significantly from the model function in eqn. (5), the calculated value for Ef may significantly over- or underestimate the explosion’s actual kinetic energy. Furthermore, the parameter k in the velocity model (Eqn. 5) will have a significant impact on the calculated kinetic energies. In table 1 k is taken to be 300; a lower value of k will yield lower Ef. A more sophisticated measurement system (e.g. a phase Doppler particle analyzer) for determining a fog explosion’s velocity and mass distributions would be needed to calculate Ef with greater accuracy and eliminate the uncertainty surrounding k.

So, if water explosions do in fact release energy, where does this energy come from? I can't yet give you a definitive answer, but I have a hunch that the energy of a water explosion comes from the reorganization of the water’s hydrogen bonding network during the explosion process. Inelastic neutron scattering experiments have revealed the existence of two different kinds of hydrogen bonds in water, each bond with a different strength (Li and Ross, 1993). The “strong” and “weak” bonds have vibrational force constants of 32 meV and 24 meV, respectively, and exist in water in a strong-to-weak ratio of about 2:1 (Graneau, 1998). During an explosion event, forces associated with the high-current arc discharge tear the water into fog droplets. This consumes a certain amount of mechanical energy, which is stored as additional surface tension energy in the fog. However, the molecules in the fog droplets now have significantly fewer neighbors than they did in the bulk water (see my cheesy little diagram below) and can more easily reorient themselves into lower energy states. Thus, weak hydrogen bonds can reform into strong hydrogen bonds, and van der Waals attractions can collapse into weak hydrogen bonds. This bond reformation would, of course, release energy, and could conceivably supply the kinetic energy of a water explosion.

For a water explosion to release potential bond energy, as described above, the post-explosion fog must be in a lower-energy state (i.e. contain stronger intermolecular bonds) than the pre-explosion water. When the fog reforms into bulk water, it must therefore absorb energy from its surroundings to restore itself to its original higher-energy state. In other words, the fog must absorb heat as it agglomerates back into bulk water (this is rather unusual, as condensation is generally an exothermic process).

In light of this, I set out to determine the amount of heat (if any) absorbed by a quantity of post-explosion fog as it agglomerates back into bulk water. To do this, I strapped a balsa wood absorber over the muzzle of the accelerator barrel. The absorber would capture most of the water expelled by an explosion in the barrel. Before an explosion event, I would measure the temperatures of the water and balsa wood absorber , and after an explosion, I would measure the final temperature of the water—balsa-wood system by inserting a temperature probe into the absorber. Knowing these temperatures and the heat capacities of the water, balsa wood, and temperature probe, I could calculate and compare the initial and final thermal energies of the water—balsa-wood—temperature-probe system , revealing whether any heat was absorbed by the water after the explosion. In these calculations, I tried to account for the resistive heating that takes place in the water arc during an explosion; I calculated this heat using the current discharge's pulse width and underdamped ringing frequency.
Table 2 below displays the thermal energy measurements for several water explosions (click to enlarge).

Table 2: The initial and final thermal energy data for 5 shots using distilled water at room temperature.  The charging capacitance was 0.6 μF for all shots.
Key: mw: mass of the water involved in the heat exchange; Tw: initial water temperature; mb: mass of the balsa wood involved in the heat exchange; Tb: initial balsa wood temperature; Tf: final temperature of the water—balsa-wood—temperature-probe system (“the system”); QR: ohmic heat accrued by the water during the capacitor discharge; Qi: initial thermal energy of the system; Qf: final thermal energy of the system; ΔQ = Qi – Qf.

Looking at the last three columns, one can see that the post-explosion fog does indeed absorb energy. For each shot measured, the thermal energy of the water—balsa-wood—temperature-probe system decreased significantly over the course of the explosion event. The difference between the initial and final thermal energies was about 75 J on average, which is sufficiently large to account for the explosion kinetic energies reported in Table 1. Of course, since not all the exploded fog was captured in the balsa wood absorber in any of the shots summarized in Table 2, each reported value for ΔQ underestimates the total heat absorbed by the fog upon reforming into water, perhaps quite significantly. So, while Table 2 tells us that the fog does absorb heat as it reforms into bulk water, it can't really tell us how much heat is absorbed in total.

It is possible that evaporative cooling is partially responsible for the low values of Tf seen in Table 2. However, I imagine the evaporation of water from the balsa wood absorber was negligible while data was being collected since the absorber’s mass did not decrease significantly while its temperature was being measured.

Now, as to this project's future: I'm contemplating another (perhaps more accurate) method of quantifying the explosions' energies other than the projectile-based method I described earlier. In the late '40s, Sir Geoffrey Ingram Taylor derived an elegant expression for the energy released in an intense explosion using properties of the blast wave the explosion creates (Taylor, 1950). Taylor used his equations to estimate the energy released in the first atomic explosion in New Mexico based on frame-by-frame images of the "ball of fire" it created. I'm hoping to use Taylor's equations similarly to estimate the energy released in a water arc explosion based on the time it takes the sound of the explosion to reach various microphones placed around the chamber. This method, unlike the projectile-based approach, would account for all the energy (light, sound, heat, etc.) released by the water explosion. I'll elaborate more on the details in my next post after I've tried this new method out.

Works Cited:

Graneau, N. (1998). The anomalous strength of cold fog explosions caused by high current water arcs. IEEE, 40/1-40/3.

Graneau, P., Graneau, N., Hathaway, G., & Hull, R. (2000). Arc-liberated chemical energy exceeds electrical input energy. Journal of Plasma Physics, 63, 115-128.

Hathaway, G., Graneau, P., & Graneau, N. (1998). Solar-energy liberation from water by electric arcs. Journal of Plasma Physics, 60(4), 775-786.

Li, J. & Ross, D. K. (1993). Evidence for two kinds of hydrogen bond in ice. Nature. doi:10.1038/365327a0

Tag, P. M. (1980, June 23). On the diagnosis and conservation of energy during condensation. Journal of the Atmospheric Sciences, 37, 2347-2351. Retrieved February 11, 2013

Taylor, G. (1950). The formation of a blast wave by a very intense explosion: I. Theoretical discussion. Proceedings of the Royal Society of London, 159-174.