OF ARTS AND SCIENCES. 173 



longer represent the proper distribution of these errors. As has been remarked, 

 they should be distributed in proportion to their influence on the clock errors. 

 The maximum periodicity' should occur at about 7'' and should decrease, slowly 

 at first, and disappear at about i^ and 10''. This is what actually happens, as 

 far as the negative residuals are concerned ; but it appears from this discussion 

 that the expression, r = m sin a + n cos a, does, as a matter of fact, represent the 

 proper distribution of the errors of single period over the Avhole 24 hours. This 

 requires that we shall find at about IS"", stars giving large positive residuals. 

 As no stars near this hour of Right Ascension appear to have irregular proper, 

 motions, the explanation so far fails. 



But it appears tliat the expression, wi sin a + n cos a, is, in a certain sense, 

 the complement of a sin d + A cos 6. An examination of the several tables will 

 show that when the corrections for errors of single period are applied, there is a 

 constant tendency to negative residuals, but that when the corrections depending 

 on the Declination are applied, the equilibrium is neai'ly restored. It therefore 

 occurred to me that the supplemental positive residuals required at IS'' may have 

 been introduced by defective or irregular pivots in some of the earlier obser- 

 vations. Inasmuch as all the Maskelyne fundamental stars which have a large 

 south declination (except Sirius) occur between 14'' and 23'^, it is required to 

 find a system of observations, which has been largely adopted as fundamental 

 in differential observations, and in which the coefiicients, a and b, are such as will 

 give large positive residuals for low southern stars. As differential observations 

 were first introduced by Pond in 1816, and as his observations have had great 

 weight in the formation of subsequent catalogues, we should naturally expect to 

 find this condition fulfilled in his observations. In order to furnish all the data 

 at present available, in deciding whether the errors in question are in any sense 

 inherent in the observations themselves, or are wholly transferred from the pro- 

 visional ijlaces of the clock stars, I give on the following pages, the residuals, 

 corrected for e, for all the catalogues at hand, between 1750 and 1871. In a few 

 cases no reduction for equinox has been made, on account of the wide range in 

 the values of the residuals. Such cases are indicated by the omission of e and 

 [v]. Whenever the first and last dates are given, and the mean epoch is not 

 given, it is to be understood that the residuals have been derived from all the 

 observations between those dates, as given in the various annual volumes. Pro« 

 visional authorities are indicated by (F). 



