The work which I have the honour of laying before the Society to-night makes no pretence to originality of idea. That credit is due to Mr. Francis Baring, Dr. Round, and their predecessors. But I do venture to claim for it some small novelty in the application to the problem of more scientific methods than have as yet been applied to it, which should yield more certain conclusions. The starting-point of this study was the attempt of the late Mr. Baring to trace the route of the Conqueror's army in 1066 by the records of the devastation which are preserved in Domesday Book. For certain counties he took the values of the manors in King Edward's day (T. R. E. valet), a date which we may call 1065/6; and also the value when the manor was received by a Norman baron—in Domesday phrase the Quando Receptum or Q. R. valet—generally taken as 1067/8. From what may be called a naked-eye comparison of these two values, he inferred that a part at least of William's army had passed through those manors which showed a heavy depreciation in 1067/8 as compared with 1066, and he provisionally indicated the routes followed by the columns. The weakness of his case, however, lay in the facts, first, that he apparently considered only the manors, and not all the holdings in each village; secondly, that he merely considered the absolute figures as recorded by Domesday. To those who were not so familiar with the Great Survey as was Mr. Baring, the method was unconvincing, and, in consequence, many scholars have remained sceptical of his conclusions.