"I was once criticized by students for amending an example I had given them when I came to work it myself. Riemann did the same thing in a big way when he changed the domain and range of implicit functions from the complex plane to Riemann surfaces. Over time a great edifice was built over Riemann surfaces while the original topic stagnated. This may be because to treat it, to understand branches, one had to use cuts which may have been regarded as arbitrary and inelegant. And if by some means branches could be got without using cuts the arbitrariness would still be there because cuts and boundaries of branches image each other. To use or not use cuts is not just a matter of taste. Previous attempts to understand branches may have failed because of a mistaken assumption that a given branch-point has a unique permutation on branches associated with it. One has first to introduce a complete system of cuts and only then for each cut determine an associated permutation. While the need for cuts cannot be got round there is a way to make their arbitrariness acceptable and to complete the theory. This is shown in detail in the other files. Copyright July2004 conesetter." The above is a quotation from a website where I used to post my notes on this topic as and when I wrote them. Then my ISP altered its software and informed me that my software was no longer compatable with theirs. So the hosting came to an end.