Dividing a circle into areas lines (there are two points on one side and one on the other). So the recurrence can be expressed as which can be easily reduced to using the sums of the first formula_3 natural numbers and the first formula_3 squares, this combines to Finally which yields The lemma asserts that the number of regions is maximal if all "inner" intersections of chords are simple (exactly two chords pass through each point of intersection in the interior). This will be the case if the points on the circle are chosen "in general position". Under this assumption of "generic intersection", the number
Circle line (London Underground) Circle Stock, at first painted red and cream, later painted red all over to reduce costs. These trains included first-class accommodation, but this was downgraded in 1940. From 1947 these were replaced by 5-car trains of O and P Stock, with doors remotely operated by the guard, released by the transfer of F Stock to the Uxbridge line. The 1933 London Underground Beck map shows a Metropolitan line north of and stations and a District line south of these points. On the 1947 map the Metropolitan and District lines were shown together in the same colour and two years later