We have two ways to think about convergency, and the book shows both — Page 247, Lesson 226

I want to walk you through a worked example that shows how we apply convergency in practice. This is a key skill for great-circle navigation, and we'll go through it step by step. We have two ways to think about convergency, and the book shows both. The first definition is: convergency equals the change in great-circle track between two meridians. The second definition is: convergency equals the angle of inclination between two meridians. Both give the same answer; it's just a matter of which approach you prefer. Let's start with the first definition. We have points A and B. The great-circle track has changed by 8.5 degrees between them. The aircraft left A on a track of 060 degrees true. Now, in the Northern hemisphere, when you travel eastwards, the track increases. That's a rule from the D-I-I-D diagram you've seen in chapter 2. So the track angle as the aircraft passes through B will be 060 plus 8.5, which is 068.5 degrees true. To find the track from B back to A, measured at B, you take the reciprocal of 068.5, which is 248.5 degrees true. Now let's look at the second definition, where convergency is the angle of inclination between two meridians. We start by paralleling the meridian at A over to B. That means we draw a line through B that is parallel to the meridian at A. The angle M is 8.5 degrees — that's the inclination between the two meridians. Angle N is 060 degrees, which comes from corresponding angles between meridians in classical geometry. So the total of M plus N is 068.5 degrees. The meridian at B defines True North at that point, so the track angle of the continuation of line AB at B is 068.5 degrees true. Therefore, the great-circle bearing of A from B, measured at B, is 248.5 degrees true. Same answer. Now let's move to Question 2. We have point C at 36 degrees north, 015 degrees east, and point D at latitude 42 degrees north. The initial great-circle track from C to D is 300 degrees true, and the final great-circle track at D is 295 degrees true. Part (a) asks: what is the longitude of D? Part (b) asks: what is the approximate great-circle track direction at longitude 011 degrees east? Let's work through the answer. The track has changed from 300 to 295 degrees true, so the convergence must be 5 degrees. C is at 36 degrees north, D is at 42 degrees north, so the mean latitude is 39 degrees north. Now we substitute into the convergency equation: convergency equals change in longitude multiplied by the sine of the mean latitude. So 5 degrees equals change in longitude times sine of 39 degrees. Rearranging, change in longitude equals 5 divided by sine of 39 degrees. That gives us 8 degrees. If the change in longitude is 8 degrees, and C is at 015 degrees east, and D is west of C — because the track is decreasing as we go west — then the longitude of D is 015 east minus 8 degrees, which is 007 degrees east. So the answer to part (a) is 00700 east. For part (b), longitude 011 degrees east is halfway between 015 east and 007 east. Therefore, the great-circle track will be halfway between 300 and 295 degrees true, which is 297.5 degrees true. Now Question 3. This time we're in the Southern hemisphere. Point H is at 40 degrees south, 170 degrees west. Point G is at 45 degrees south, 174 degrees east. The initial great-circle track from H to G is 250 degrees true. The question asks: what is the initial great-circle track from G to H? We draw an initial great-circle track of 250 degrees true. In the Southern hemisphere, the track must be from the right-hand side of the diagram if it's going to cut the other meridian. So the right-hand meridian must be H, and the left-hand one must be G. Let's double-check: H is at 40 south, G is at 45 south, so G is south of H on our diagram — that's correct. H is at 170 west, G is at 174 east. We're crossing the Greenwich anti-meridian, which is 180 degrees east/west. Left and right are the right way round on this diagram, which can be confusing, but it's correct. If you need clarification, check the explanation in chapter 1. Now we use the convergency equation again: convergency equals change in longitude times sine of the mean latitude. We'll work through the numbers to find the track from G to H.