21
$\begingroup$

Your miners find some strange stones - they float to the ceiling of mines.

With those stones you can easily build an airship - just fill a sleek wooden hull with them and give the rowers some wing-paddles.

With a drag coefficient ~0.02, even under muscle power you can get some tens km/h, also you can use favorable wind.

Now you have one problem - there is no way to regulate lifting power. If you let the floating stone go it will go straight to space.

And the ship is constantly losing weight as rowers consume food and water. Also accidents happen - a paddle gets dropped, a rower goes overboard, some barrel gets broken and spills wine. Or you want to drop some flechettes on enemies' heads. And rowers can provide only so much thrust downwards...

You can just let some stones go - but they are very expensive. You can gather rainwater or dew from clouds - but that depends on weather.

Dropping ballast allows you to make the ship lighter. But you need antiballast - a way to make ship heavier in the middle of flight.

So what is an optimal way to increase the weight of, say, a 10 ton ship by at least 100 kg - I guess some reaction to bind oxygen or vapor that is accessible without industrial technology?

Update.

Compression of air is not a very optimal way to do it. Even with modern technology, a balloon weights several times more than air it contains. For preindustial one ratio would be order of magnitude worse.

Likewise, cool air - suppose we have a 10 ton 100 m3 airship with 300K on inside. If we cool it even to -70C then we would get about -30 kg, less than weight of adult. And making the ship bigger would increase drag and make it too slow under manual power.

$\endgroup$
8
  • 5
    $\begingroup$ Just FYI rowing would barely work and would be incredibly difficult to do even badly efficiently. You'd just have a propellor on a stick, and many hands would spin the stick. $\endgroup$
    – Fattie
    Aug 7 at 0:03
  • 6
    $\begingroup$ FYI, rowers consuming food and water wouldn’t change the weight, unless they disposed of their bodily waste outside the ship. However, being alive will reduce the total weight (or rather the process of converting o2 into co2 and letting that co2 fly away) $\endgroup$ Aug 7 at 9:02
  • $\begingroup$ Dammit, this stuff just asks for a space lift. Imagine a chain/rope of steel intertwined with these stones. Mass X, weight 0 (adjustable at making, if you want, make it negative weight), scalable tension to allow lifts to accelerate vs the thing, AND it has enough inertial mass to not break in an instant when employed! Aaaaawwwww... $\endgroup$
    – Vesper
    Aug 8 at 7:48
  • $\begingroup$ You should also think about how the ship takes off and lands. Keep in mind that the same constant force that makes it take off will (if left unchecked) send your ship to space. $\endgroup$
    – Blueriver
    Aug 8 at 15:56
  • 1
    $\begingroup$ See how real-world airships did it: hydrogen is expensive, helium is very expensive, and burning fuel causes airships to lose weight. $\endgroup$
    – Mark
    2 days ago

18 Answers 18

34
$\begingroup$

Volume-based Adjustable Buoyancy

The key here is to not use so many stones that your ships float up. Instead, use as many stones as is needed to make your ship effectively weightless. If 1kg of floating stone(determined by upside-down scale) makes 1kg of normal matter weightless(not falling, not rising)(just guessing, you've left little as far as hard numbers are concerned) then 2.5tons of a 5ton ship will need to consist entirely of floating stone

Doing so will make it possible for you to use an already tried-and-true method of altitude adjustment, namely a hot air balloon of some sort, whose own volume and lift will determine whether the whole of the ship goes up or down. As crew and cargo is loaded the weightless ship will naturally increase in weight and drop/sink, but with the ship's weight itself being accounted for by the floating stone matter you should be able to get away with relying on the balloon alone for lifting both crew and cargo, with the burners supplying the heated air and their fuel being the prime determiner of how far/long a ship can go without getting landlocked(unable to float) and may provide a nice niche for the economy to flourish in. They can also land and have the crew chop some wood for fuel instead of using coal if they're desperate.

Alternatively

You could still go with the weightless ship tactic and do a small amount of hand-waving to say that they've discovered a way to use wooden gears + wooden fanblades + manpower to produce a primitive propeller and engine of sorts whose constant additional lift in addition to the lift from the floating stone provides enough power to move/lift the ship along with its crew and cargo, with going down being as simple as telling the crank crew to slow down the cranking of their cranks(the cranks in this case having replaced the function of oars, with part of the crew being in charge of ship lift and the rest being in charge of ship thrust(if you want a sail-less option or the wind is against you))

$\endgroup$
6
  • 4
    $\begingroup$ clearly, (-)1kg floating stone will counteract exactly 1kg of ordinary mass. per definition, if you use a "upside down scale") $\endgroup$
    – ths
    Aug 6 at 17:22
  • 1
    $\begingroup$ @ths I was initially going to reply on how I simply ran with a more favourable ratio as otherwise it's not much of a fantasy, but then I thought about it and realized what you meant with your comment. Of course 1kg of negative mass(floating stone) will only counteract 1kg of normal mass. Although this does reveal my mind was more on density volumes than pure mass values(denser materials being smaller volume-wise than less dense materials when both materials have the same mass, hence the 1-5 assumed ratio) - Editing to correct $\endgroup$
    – Lemming
    Aug 6 at 19:19
  • 2
    $\begingroup$ I guess these floating stones still have a positive inertial mass, just a negative weight. $\endgroup$ Aug 6 at 23:07
  • 1
    $\begingroup$ That's clever ! $\endgroup$
    – Fattie
    Aug 7 at 0:02
  • 3
    $\begingroup$ Extending on the hot air balloon idea: Perhaps the magic stones provide more "buoyancy" when heated up, and thus you don't need the large and vulnerable gas bags. $\endgroup$
    – Simppa
    Aug 7 at 9:46
26
$\begingroup$

Harvest extra ballast water from clouds.

The Graf Zeppelin did this in real life during the 1930s. A set of gutters on the side of the vessel collected the water. The airship brushed against the edge of a cloud or light rain to harvest the water.

In real life, opportunities to do this were somewhat uncommon and irregular -- weather can be unpredictable. Trip planning never counted upon rainwater collection; it was a bonus that conserved expensive hydrogen. But perhaps conditions are slightly more favorable on your world.

(Oh, don't dive into the cloud or rain. The zeppelin officers, mostly veterans of WWI who had lived through hydrogen fires and airship crashes and faced down incendiary machine guns and the bitter North Sea, who had navigated into combat by guesswork upon cloudy nights and seen most of their comrades die...those adrenaline-addicted adventurers considered diving INTO a cloud or rain far too risky to attempt.)

$\endgroup$
1
  • 2
    $\begingroup$ Holy! who knew? $\endgroup$
    – Fattie
    Aug 7 at 0:03
14
$\begingroup$

Build a chain which incorporates some of the floating material. The resulting chain will be "weightless" and can therefore be of arbitrary length - instead of ultimately ripping under its own weight.

Whenever you need more weight, lower a few people down to the ground, let them shovel some dirt into a container and pull everything back up.

$\endgroup$
4
  • $\begingroup$ Very smart, did not think about it! Though it still needs some weight to go down despite wind and it needs to withstand wind load and weight of dirt, so it restricts the usability of the method. But we can use it to deploy sail in the layer of air with different speed. $\endgroup$
    – Vashu
    Aug 7 at 23:38
  • 2
    $\begingroup$ @Vashu just attach an anchor to the chain. The trick with weightless chain is to make it not break up under its own weight, yet any weight attached would act as normal. $\endgroup$
    – Vesper
    Aug 8 at 7:46
  • 1
    $\begingroup$ Even though the chain is weightless, it still has mass and therefore cannot be of arbitrary length, unless you want to spend an arbitrary amount of time reeling it in. F=ma is inescapable - the maximum value of F is the breaking strength of the chain, and m increases as the chain gets longer, so arbitrarily high m implies arbitrarily low a. It's like pulling a heavy object on a horizontal frictionless surface - you can pull hard with a thick rope, but even a weak tug would snap a thread, even though gravity plays no role whatsoever in this scenario. $\endgroup$ Aug 8 at 14:22
  • 1
    $\begingroup$ @NuclearHoagie Well, I don't see this as a serious obstacle, and yes, extending such a chain would take arbitrary time, and indeed, using only the anchor's weight to unfold a mile long chain with inertial mass a hundredfold would take quite a while. Still the underlying idea would work. $\endgroup$
    – Vesper
    Aug 8 at 18:48
13
$\begingroup$

The stones are expensive, so you dont add so many that the ship becomes weightless.

You essentially build a Heavier-Than-Air airship. It will naturally float down if nothing is done, but by the use of wing surfaces, fans, propellors and even oars designed as fans you can gain the lift required to gain altitude (use only one or a combination).

$\endgroup$
2
  • 2
    $\begingroup$ Note that this is how the vast majority of real-world airships work: the airship doesn't carry enough hydrogen or helium to float; rather, it carries almost enough and uses the body of the airship as a giant low-efficiency wing to make up the difference. $\endgroup$
    – Mark
    2 days ago
  • $\begingroup$ Its also how flying animals work. They use work and aerodynamics to bridge the gap between walking on the ground and flying through the air. To be fair there are microcreatures that have the right Stokes number to not have to do as much. Just like a small enough grain of dust is entrained and carried away by the wind, there are mites that also get carried like that. $\endgroup$ 2 days ago
9
$\begingroup$

All matter has mass. The stones are in the mountain because when the mountain was molten, and the material comprising the stones was still there, they did not float away into space.

There is a material in the real world called "sol-gel foam" that is really a very sparse ceramic mesh. Folks have pulled vacuum on it, and coated it in plastic, and it is positively buoyant in air. It is lighter than sea-level air; it acts a lot like helium. It does not repel from normal matter, and wouldn't float all the way up to outer space. It would find its level and try to float there.

A submarine can make itself more dense, and sink, by taking in mass. An airship could take on air, and compress it, and gain mass, and therefore weight, and adjust buoyancy in that manner. There would still be a "local density" based on pressure and temperature to determine how much lift the material provides. (descriptive link)

A hot air balloon gets lower density by heating up air. (descriptive link) If you want more density, then cool it off. You could have cryogen canisters to make "anti-balloons" that pull the ship down. You could also get mileage with propellers.

The Hero-style steam engine (greek) suggests ancient Greek steampunk was wanting a Newton. In your story, you could supply one. Imagine Alexander the great with a railroad, or decent gatling guns. They could also have Da Vinci style air-screws, or necro-robotic wings. Spiders aren't the only Arthropods, so you could have trilobites, butterflies, scorpions, centipedes, and crabs as candidate articulators. The Meganeura Meganisoptera/dragonfly had a wingspan of 2-feet, suggesting insects that operate at least in the envelope of modern birds.

A fun feature of aerogels is they are superb insulators. You would not get a Hindenberg Zeppelin event there. You could bounce flares off it all day long.

Bibliography:

  • Necro-robotic articulators at Rice (link).
  • Lightweight/buoyant aerogels (have air inside) (link, link)
$\endgroup$
2
  • 1
    $\begingroup$ I wish I could give an extra up vote just for the "necro-robotic" wings suggestion. 😂 $\endgroup$ Aug 7 at 7:06
  • 4
    $\begingroup$ This is the simplest, and most physically plausible solution - they don't have negative density, they're just less dense than air. They won't go all the way to space, they'll reach an equilibrium at some height. $\endgroup$
    – kaya3
    Aug 7 at 18:31
6
$\begingroup$

Heat the stones. Whatever handwavium takes place to make the stones generate antigravity is reduced when the stones are heated. The Handwavium returns as the stones cool.

Bonus, that is why the stone is so rare. Most of the stones have "evaporated" from the planet because during the formation of the planet if they were near the surface and cooled they just floated away.

$\endgroup$
6
$\begingroup$

I'm surprised that nobody else has suggested this. Don't rely 100% on floatium for your buoyancy. Have a hot air balloon that takes your ship over the threshold. You don't need huge blimp-sized balloons with this solution. You can probably get away with one smaller than the ship itself. You just need one big enough to compensate for your live load: the passengers, cargo, etc. that you don't keep on the ship most of the time.

It occurs to me that you would need floatium pallets with the cargo, regardless, because otherwise the ship wouldn't be able to stay on the ground when unloaded.

This connundrum will actually make it difficult to create floatium war ships. Every time you fire a cannonball, you'd have to adjust your buoyancy to match.

$\endgroup$
5
$\begingroup$

I guess we could use reaction of slaked lime with carbon dioxide.

Ca(OH)2 + CO2 → CaCO3 + H2O

But to get 1 kg of weight this way we would need to pass at least 3 ton of air through lime-water. Water vapor loss could negate the effect.

Another way would be to corrode iron fillings

4 Fe + 3 O2 + 6 H2O → 4 Fe3+ + 12 OH− → 4 Fe(OH)3 or 4 FeO(OH) + 4 H2O

But it is hard to do it fast, especially if we do not add water. And if we add water then efficiency is only 29%.

Burning steel wool is fast and efficient, but it was invented in 1896, so it could be hard to produce it in pre-industrial times. Same for burning magnesium.

I guess burning iron fillings / lead, antimony or arsenic is the best option.

$\endgroup$
4
  • 1
    $\begingroup$ Can you explain why is "Harvest extra ballast water from clouds" the only relevant answer so far? There are a number of other suggestions in the other answers. Any additional limitations should be added to the question so that any subsequent answers could take these into account. $\endgroup$
    – rooby
    Aug 8 at 4:03
  • $\begingroup$ @rooby Third of them is about downward thrust - that was mentioned in question, third about cooling air - that I can add to question (not really effective). Answers like compression of air are not very effective without high technology and multi km chain is not really realistic. $\endgroup$
    – Vashu
    Aug 8 at 6:55
  • $\begingroup$ @rooby Anyway, I guess I should not be so strict so I dropped "unrelevant" stuff. $\endgroup$
    – Vashu
    Aug 8 at 7:08
  • $\begingroup$ I second the idea of burning iron filings. Iron is pretty cheap and it should be pretty easy to burn it. The ashes could even be smelted back into iron later (but probably not during the flight). $\endgroup$ Aug 8 at 12:25
5
$\begingroup$
  1. How about using guide ropes?

They pretty much self-regulate to a great extent.

https://en.wiktionary.org/wiki/guide_rope

  1. Make the whole thing a hot air balloon.

Your magic stones will make it extremely efficient, requiring 2% or 5% of the lift to be hot air. This will make your aerodynamics much better than a traditional hot air balloon, but you can still release the hot air to an extent or simply turn down the fire and sink to the ground.

Of course, you are free to combine both technologies.

$\endgroup$
1
  • $\begingroup$ I was thinking of the hybred solution too. First of all combining a hot air balloon and magic rocks allows you to use less rocks, and secondly you can reduce the lift just by not heating the air (just like a hot air balloon alone) The rocks basically allow you to make HUGE hot air balloons, and you can make them out of as heavy a material as you like (with enough stones to counter the weight). Since the whole thing still has mass, inertia could be a bitch. $\endgroup$
    – Bill K
    Aug 8 at 20:47
4
$\begingroup$

Ordinary, unheated air.

At jet cruising altitude, outside air is -40 C to -70 C = 203 to 233 K. Comfortable cabin air = 298 K. 218/298 = 73%. Air inside cabin weighs more than 1/4 less than outside air.

Turn off heat. Open all windows. Wear coats - optional, rowing is hard work. Wear oxygen masks - not optional. (Alchemy? Recover from burnt steel wool? Does this planet have nitrogen atmosphere?) Assuming penteconter construction, 30 m x 4 m x 3 m = 360 000 L = 15 000 mol = 470 000 g x 0.37 = 170 kg of mass enters through open windows.

$\endgroup$
3
$\begingroup$

Birds

Dump some bird seed on the deck and play some territorial bird calls. Suddenly every bird in a kilometer radius has landed on your air ship and is weighing it down. You will have to be very careful not to startle the birds though because after they finish eating your bird seed they may take off again making your ship even lighter than it was. But hopefully they can weigh you down enough that you can hook onto a tree or something-- giving your crew the opportunity to go out and secure more bird seed for future landings. If you do this frequently enough maybe flocks of birds start following your ship around waiting for you to give the signal that you're about to dish out more food.

$\endgroup$
2
$\begingroup$

Reduce lift by using a tethered "balloon" to temporarily eliminate the lift of the stones.

Simply throwing the stones overboard is not feasible due to their cost, but you can throw them overboard if you have a means of retrieving them. Store your lift stones in a crate attached to deck via a long rope wrapped around a winch. In the static configuration, the crate provides lift to the ship. By releasing the winch, however, the stones lift the crate alone, and provide no lift at all to the ship. As long as the rope unspools, the ship will descend, although you'll be in the exact same situation of "too much lift" when the rope is completely unspooled. You just need a rope long enough to allow the ship to descend to the ground, at which point it can be tethered down, allowing the winch to be spooled back up.

This wouldn't be well-suited to continuous fine adjustment of the amount of lift, since it only provides a temporary reduction in lift force, and requires landing to actually change the balance of the ship. It could be useful as an emergency measure, however, similar to how a submarine can surface in an emergency by blowing the ballast tanks. This is effectively the opposite, allowing the airship to descend rapidly in an emergency, and mitigates the cost of doing so by allowing the lift stones to be recovered.

$\endgroup$
3
  • $\begingroup$ at the extreme, you have the entire ship dangling by a (magic) rope underneath the entire lifting section and you let the lifting section rise as much as it wants to, while paying out rope to keep the ship at the target height. Doesn't seem all that useful! $\endgroup$
    – user253751
    Aug 8 at 18:11
  • 2
    $\begingroup$ @user253751 The goal of this system is to get to the ground and fix the lift issue there, not to maintain a target altitude, as mentioned in the second paragraph. $\endgroup$ Aug 8 at 18:49
  • $\begingroup$ Could be an interesting mechanic for dodging enemy fire. Just make the whole ship quickly drop down if you see any incoming cannonballs. $\endgroup$
    – Dugan
    2 days ago
0
$\begingroup$

Those stones don't "float" (which just means they are less dense than the surrounding medium, and that there's a maximum service ceiling). No, these are really weird stones: the provide a constant, directed force without requiring power. If directed upwards, they start rising, until the cave ceiling stops them. If directed downwards, they start descending (faster than with just gravity), until the floor stops them.

So the solution is to mount them to your ship on a turnable bearing. Mount them in pairs that turn in opposite directions - when you point them at each other, the force cancels out, and gravity takes over.

$\endgroup$
0
$\begingroup$

Use a large stone tethered to a long cable spool on a drum. Release the drum and allow the airship to drop whilst the stone rises. Just before the stone reaches the end of the tether release large hinged flaps that drop down to increase the surface area of the ship.

When the stone reaches the top of the tether the ship receives a big pull upward, but a lot of the energy of the upward pull is wasted against the increased air resistance of the flaps. As the ship starts to rise lower the stone on the tether with the flaps out then pull the flaps in and release the stone again.

$\endgroup$
0
$\begingroup$

According to good ol Wikipedia, silica gel has been around since the 1640s, and was used in WWI to absorb vapors in gas masks. (https://en.wikipedia.org/wiki/Silica_gel) It can absorb about 40% of its weight in water without changing their size. This moisture is released when the gel is heated up.

Hydrogel crystals can absorb anywhere from 500 to 1500 times their weight in water (https://www.nature-and-garden.com/gardening/hydrogel.html), and can be made from starch, cellulose, or petroleum. I am not sure if each of these require industrial technology for production.

Another potential solution could be a basket filled with absorbing fibers. The dry basket could be lowered into a lake, river, or ocean to absorb water and then be hauled aboard with the added water weight. Heck, even the bucket could work without the fiber. A gallon of water weighs around 8.3 lbs./3.78 kg. A 15-gallon bucket of water would weigh around 125 lbs. A 50-gallon bucket would weigh around 415 lbs./204 kg. Winch down a bucket, and pull up the water needed to increase your weight.

Adding a source of cold, such as blocks of ice, would cause condensation from the air to form. This could provide a source of additional water without needing to fly through clouds. The ice could be kept insulated until it was needed for condensation capture. There are Dew Point Calculators (https://www.calculator.net/dew-point-calculator.html?airtemperature=70&airtemperatureunit=fahrenheit&humidity=40&dewpoint=&dewpointunit=fahrenheit&x=92&y=23) which are helpful in determining the temperature difference needed to cause condensation to form based on the relative humidity level. The rate of condensation formation is based on the amount of airflow around the condensation point, and the relative humidity in the air. I was not able to find any good calculations for the rate, likely due to the complexity of the subject.

You might be able to do some experiments where you weigh ice cubes before and after they melt while in front of a fan. If you know the relative humidity inside your house, you can time the rate at which the water increased in weight by dividing it by the time it took for the ice to melt. Be sure to include any condensation on the outside of the container as well. “End weight – start weight / time”. It would be a rough estimate, but if there is a lack of a true formula for calculating the condensation rate, nobody will be able to argue.

It would be nice to know if the stones are simply lighter than most other substances or exhibiting anti-gravity properties. If they are simply lighter, then there would be an altitude zone where these stones would hover with neutral buoyancy. As the number of stones in the upper atmosphere increased over time, they could cause global cooling. If they exhibited anti-gravity, then they would leave orbit and head towards deep space.

$\endgroup$
0
$\begingroup$

Aerodynamics can handle this, like blimps do using the lift of their elongated body and tail control surfaces.

If using pre industrial technology, wood and fabric windmills-like propellers, wooden gears and pulleys, wooden shafts, leather belts, powered by arms or legs, could allow to give a vehicle the required speed to be aerodynamically controlled. This vehicle does not need a baloon or enveloppe, just horizontal and vertical surfaces, and a cabin for people to do fitness.

The vehicle has to be slightly heavier than what the stones can lift, the rest is aerodynamic human powered lift.

$\endgroup$
-1
$\begingroup$

Floatium is contained in a rotating drum

Depending how much negative force the Floatium induces (does it rise inverse to gravity?) a drum rotated by sailors can be used to induce a tumble effect and "toss" the Floatium and losing contact with the barrel surface. While the Floatium is out of contact with the drums surface and in the "dead space", it's upward force isn't applied to the ship.

  1. Barrel rotates
  2. Floatium loses rotational momentum and rises
  3. Floatium out of contact with surface

These barrels would need to be used in counter-rotating pairs so not to impart roll.

enter image description here

$\endgroup$
6
  • 3
    $\begingroup$ Even with reverse gravity physics, this would be a perpetual motion machine. $\endgroup$ Aug 8 at 23:27
  • $\begingroup$ Modified—there’s men or machines rotating the barrels with the sole purpose of getting floatium off the surface. This probably induces a climbing effect as well but physics isn’t my strong suit. $\endgroup$ 2 days ago
  • 4
    $\begingroup$ Between phases 1 and 2 the floatium raises the ship AND the drum "climbs" upwards as well on the stones, therefore balancing out the freefall phase of 3. $\endgroup$
    – zovits
    2 days ago
  • 1
    $\begingroup$ @Dugan The stones are constantly trying to move upwards. If the drum is rotating so, then it exerts additional upwards force on the airship. So the stones are falling 50% of the time with 0 net force and are being rotated down 50% of the time - exerting twice the force. Like climbing up on a rope that's being pulled up the same time. $\endgroup$
    – zovits
    2 days ago
  • 4
    $\begingroup$ As the drum pushes the Floatium down in Step 1, the Floatium pushes the drum up. When the Floatium hits the top of drum in Step 3, the whole drum moves upward. The zero force imparted in Step 2 is averaged out by the increased lift force in Steps 1 and 3. Turn the whole system upside down with regular gravity, and it supposes that you can reduce the average apparent weight of a bucket simply by raising and lowering it on a pulley, but no such system exists - such a system would be widespread to reduce payload weight for flight or rocket launches, allowing launch of arbitrarily heavy payloads. $\endgroup$ 2 days ago
-1
$\begingroup$

I'm surprised nobody else mentioned this, but:

Consider pulverizing the stones. For real, make a fine dust out of em and fill that in your airship balloon or w/e you're using for stone storage. This way, you can have a ventilation to let off JUST the amount to sink and don't waste any.

Or maybe just dump the idea with the stones after all, it's garbage. The zeppelins of old times where dumped too because they where simply inefficient. In a world with magical floating stones, somebody has to come up with a better solution than copying a real-world airship.

$\endgroup$
2
  • 2
    $\begingroup$ Zeppelins are inefficient because they require a huge amount of expensive gas which inevitably leaks out and needs to be replaced, as well as a large, non-aerodynamic envelope that limits speed. Neither the aerodynamics nor the consumable nature of the lift are an issue here (unless you throw the stones overboard as you suggest, which the OP explicitly wants to avoid). I don't see any particular reason why an aircraft using lift stones would need to look anything like a zeppelin. $\endgroup$ 2 days ago
  • $\begingroup$ The other big problem with zeppelins is the size of their support facilities. Their hangers were some of the largest buildings of their time, while an airship port needs a quarter of a square kilometer of land for each Hindenburg-sized airship moored. $\endgroup$
    – Mark
    2 days ago

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .