sago

This is annoying because I know I should be able to do this, but I can't see how.

I have a cable hanging between two anchor points at different heights. I know: the distance between the points, the height of each point, and the height of the lowest point in the sag.

I want to find the characteristic value of the catenary (normally called a).

Unlike the solutions around on the internet, I _do not_ know the length of the rope!

For the sake of making the problem simpler, I assume the lowest point of the sag is at zero height, and rescale the heights accordingly.

So I know y1 (height of anchor 1 above lowest point of sag), y2 and x (total distance between anchors).

Assuming that the cable touches y=0 at x=0, I get three equations:

y1 = a cosh(x1/a) - a

y2 = a cosh(x2/a) - a

x = x1 + x2

In three unknowns: x1, x2, and a

And then I stare at my paper with bottom lip quivering.

Anybody feeling brave?

I'm not interested in any packages or software, just how this is done conceptually.

If I bill European clients in euros where I normally use US$, how does the exchange rate work when putting together a transaction.

For example: I invoice Company A for E2000. I get a transaction like

Sales - CR $2400

Invoices Payable (Euros) - DB E2000

Then they settle up with me

Invoices Payable (Euros) - DB E2000

Cash at Bank - CR $2200

The exchange rate changes all the time - and as it stands I've lost $200 here - how do you get these figures to balance?

I'm not an accountant, and I have magical software to actually do this for me, but I'm trying to understand how double-entry bookkeeping works, so I'd like to know :)

Thanks in advance.