Roll gaps in a continuous series

Natural gas futures (illustrative, $/mmBtu): at the end of August the October contract is at 2.90. On the roll date in late September the October contract is at 3.00 and the November contract at 3.30. At the end of September the November contract is at 3.35. (a) Compute the September log return of the continuous front-month series, the return of a position held in October and rolled into November at the roll date, and the roll gap. Verify (10.24). (b) Suppose the log curve has the stable seasonal shape of Proposition 10.14 with delivery-month prices (Jan to Dec) 3.60, 3.50, 3.25, 3.00, 3.00, 3.05, 3.10, 3.12, 3.10, 3.15, 3.35, 3.55, and the continuous series rolls in late month $m$ from delivery month $m+1$ into $m+2$ (so the September roll goes from October into November). Compute the roll gap credited to each calendar month and verify that they sum to zero. (c) Which calendar months get the largest positive and negative gaps? Compare with the NG=F month means in 10.13.4. (d) Explain why an investor who holds and rolls NG futures cannot earn the September effect in (b).