responding lengths of time. The tape was prepared for use by being marked off by printed dots, about a third of an inch apart, into spaces representing seconds. The permanent record was prepared subsequently by reading off the results noted upon the tape. To read off the whole seconds was of course a simple matter of enumeration of spaces representing seconds. The fractious of a second, however, it was customary to estimate in tenths of a second. The estimation was made upon the position of one sharply-marked dot as referred to two other well-defined dots, one on each side of it, indicating the beginning and the end of a second, and separated about a third of an inch. It was done by inspection of the tape, by highly-trained and experienced men, to whom it had been a daily work for years. Such being the methods, an astronomer connected with another observatory selected at random from the reports of the Greenwich Observatory a large number of records, and caused the number of times each fraction of a second occurred to be counted. Theoretically, there is no reason why one fraction should appear more often than another. An examination of over 1,200 instances, however, showed that certain fractions appeared much more frequently than they theoretically should. The figure 4 or 10, and the figure 0 or 10, were found too often. Upon this fact was founded a criticism upon the accuracy of the reports. It was claimed that the frequency of these fractions was occasioned by personal characteristics in the person who estimated the fraction; and it was assumed that such were the idiosyncrasies of even the most highly-trained persons, that in making such estimations they would unconsciously tend to use certain figures rather than others; in this case it was argued that the tendency causing error was to make the record four-tenths for most fractions between threetenths and five-tenths, and where the dot was near the end of a second to record the time as a whole second. This criticism was offered in a dignified and serious way in a prominent scientific journal, and was as earnestly replied to and discussed by the observers at Greenwich in the same journal.
This example of the personal equation is quite different from that which was first briefly described. The value of the equation in this case it is impossible for us to formulate with accuracy in the present state of our knowledge. In examples like the first, the factors can be more readily observed, analyzed, and measured. The difference which appeared in the case of Maskelyne and Mr. Kinnebrook arose, without doubt, from the fact that nervous and mental actions require time for their accomplishment, and because the rate of nervous transmission and mental action in one of these observers constantly differed from that in the other. The problem of the personal equation in this aspect becomes one of physiology and psychology. As such it has been investigated with great research by specialists during the past twenty years. And, although the results obtained are in most cases only