In the last number of the Journal I explained and exemplified a method for the formation of logarithms and anti-logarithms to twelve places, and gave the tables requisite for its application. I am now to analyse and farther exemplify the method in question, and demonstrate the rules laid down in the previous paper.

When a series of quantities which depend on a fixed law, whether known or implied, follow each other in a due order of succession, the general accuracy of their numerical values may be satisfactorily tested by observing the regularity of the progression of a suitable order of differences. If the tabular quantities are the results of calculation from a given formula, with equidistant arguments, by differencing them up to a certain order the existence of an isolated error, if one should exist, is thus prominently exhibited and therefore speedily detected. Moreover the precise locality of the error and the fact of its individuality being indicated by a characteristic interruption of the law of progression, the proper correction is ascertained by carefully noting the central position of the disturbance, and revising the corresponding portion of the original computation. This work of revision may, however, be dispensed with, if necessary or desirable, as the magnitude as well as the locality of the indicated error can in general be readily deduced, with sufficient accuracy, directly from the differences, by the method of adjustment here given. The method, being so far independent of fundamental origin, is therefore equally applicable to the adjustment of a series of quantities derived from observation, or from statistical data; only in this latter case, as almost every quantity must be more or less affected by imperfections arising from defective information and other known or unknown incidental causes, the influence of which may sometimes be augmented by paucity of numbers, the progression of the differences is disturbed by a more complicated combination of the effects of an irregular series of errors, and it becomes more difficult to eliminate the value of each separate correction.