 So, we've just defined this void ratio parameter. We can relate it to the parameters that we've previously defined, saturation ratio, water content and specific gravity. Remember that the water content and saturation ratio describe something to do with the water in the soil. So, we can connect these things together through a single equation, and I'm just going to go through the proof of that now. So, the first thing we do is we can take the specific gravity and write it in terms of the three phase model values. So, take the specific gravity and write these densities as masses and volumes. So, specific gravity equals the mass, so the density of the solid, which is the mass of the solid over the volume of the solid, divided by the density of water, which is, well, if you multiply, if you flip it around and multiply it, so it will be the volume of the water over the mass of the water. Okay. Okay, so if you take the specific gravity and we multiply it by the water content, we have the water content, which is the mass of the water over the mass of the solid, multiplied by the mass of the solid, and multiplied by this, so the mass of the solid over the volume of the solid, multiplied by the volume of the water over the mass of the water. You see that we've got, in this part of the equation, we have a mass of water on the top and on the bottom, and a mass of the solid on the top and on the bottom. So, the mass of the water is cancelled out, and the mass of the solid is cancelled out. What we're left with is the volume of the water over the volume of the solid. So you can see we're approaching terms on this equation that might be able to be substituted in terms of hydration. We've certainly got the volume of the solid on this side. But what we need to do is get, essentially, the volume of the water over the volume of the solid on this side. So to do that, we just divide by the saturation ratio. So if we divide this by the saturation ratio, we can multiply this side by, flipped over side of this, so the volume of the void over the volume of the water. And what we have here is the volume of the water and the volume of the water on the top and bottom. What we're left with is the volume of the void over the volume of the solid, which is, in fact, the void ratio. So we can get rid of that and write it as the void ratio. So we can also relate the parameters that describe the voids within the soil to the bulk density. This is really important when we're trying to work out what these parameters are in soils because they're particularly hard to measure. So it's really hard to measure porosity, void ratio, and specific volume in the lab. It's a lot easier to measure bulk density, water content, and specific gravity. And the equation I'm about to derive now relates these things to this specific volume. So when we're working in the lab to find out specific volume, we calculate it through this equation. OK, so the first step in relating these four parameters together is to take the inverse of the bulk density. So if we take one over bulk density, we essentially just flip this side of the equation. So we have total volume over the mass solid plus the mass of the water. And then if we multiply it by specific gravity, so we have specific gravity over bulk density, we have the total volume of the mass of the solid plus the mass of the water times by this up here, which is the mass of the solid times the volume of the water all over the volume of the solid times the mass of the water. Now what we can do in this case is collect the total volume and the volume of the solid together and then put everything else together. So what we can rewrite this in terms of the total volume over the volume of the solids multiplied by the volume of the water times the mass of the solid all over the mass of the water and the mass of the solid plus the mass of the water. So it's just rewriting that equation in terms of total volume over the volume of the solids and we can see that that is the same as specific volume. The other thing to notice is that over here we have the mass of the solid over the mass of the water, which is the same as the inverse of the mass of the water content. So that's just saying that the water content is on the lower half of this equation. So it's one over water content, so we can just pop the water content down here. And that's actually, instead of leaving it down here, we'll move it up here. So we'll just multiply both sides of this equation by the water content. So sort of putting it there, we can put it up here. OK, so we can get rid of those brackets as well. So we have four of those terms together. We just need to do something about this over here. OK, so the next step is to inverse everything again. So if we flip everything around, so we have the density now on the top, divided by the water content and the specific gravity equals one over the specific volume multiplied by the mass of the solid plus the mass of the water all over the volume of the water. We can now separate this out into two terms so we can rewrite this. OK, so we've inverse that and separated out the volume of water into two fractions. So you can see here that we've got the mass of the water over the volume of the water. So that's just the density of water. So we can rewrite this in terms of density, density of water. OK, so we're getting there, but we still have the mass of the solid over the volume of water to deal with. Now if you take the density of water and you divide it by the water content, try and work that out from using these three phase parameters. But essentially what you're left with is the mass of the solid over the volume of water. And you can work that out for yourself if you like. So we can see that we can slot this into here. So instead of mass of solid over volume of water, we can write the density of water over the water content. Actually I should put brackets around those just to make it clear. OK, so we've now got an equation that relates the bulk density, the water content, and the specific gravity to the void ratio. And there's no, none of these three phase parameters within that. So all we really need to do now is tidy up this equation. It's a little bit messy. Now the way you can do this is if you take the, if you bring the water content over to the other side of the equation now. So we've got bulk density over specific gravity equals water content over specific volume, or the density of water over the water content plus the density of water. And if we multiply this bracket by the water content, what we're left with is, well, the water content up here which cancels with this and then the water content over here. So multiplying the final density of water by the water content. So if we take, now take the density of water outside of the brackets. We now multiply this through so we can get rid of that. OK, one over V. So we take the density of water outside of the brackets. Take it here, density of water. It's now being multiplied. We can have one plus just the water content. Really, we want to write this equation in terms of just a specific volume on one side. So what we can do is multiply both sides by specific volume. So we have specific volume on one side. Divide both sides by density. So we have, sorry, multiply both sides by specific gravity first. So you have specific gravity, one plus. And then divide both sides by the bulk density. So we have density of water over the bulk density. And this is the equation that's really quite useful. It's in a form that relates bulk density, water content, and specific gravity to specific volume. So this is easy to measure in the lab. This is easy to measure in the lab, and this is easy to measure in the lab. So we can work out from parameters that are easy to measure, the parameter to do with the voids within the soil. And if we know one of those, we can work out void ratio and porosity.