OP ARTS AND SCIENCES. 173 



"We may still further introduce the term D c?,, representing the diur- 

 nal aberration, giving the form (e), and by substituting R. A. -}- E e 

 for R. A. where E e represents a term depending on 2 D , we get 

 the form (/). 



Finally, if we represent by the constant the personal equation 

 between bright and faint wires, bright and faint stars, &c, we have 

 the form (g). 



Of course it is wholly absurd to introduce the terms D d, E e, and 

 C as unknown quantities, and these forms are given only to show that 

 one must exercise sound judgment in the formation of the equations 

 in order that the solution by least squares shall give correct results. 

 It is useless to expect that the solution will separate errors which 

 appertain to different instruments. For example, in form (g) it would 

 seem hardly necessary to say that the solution will entirely fail in 

 assigning to the telescope the correct values of a, b, and c ; to the clock, 

 the true values of AT and x h, to yield the physical constant which 

 enters into the diurnal aberration, and the coefficient which results 

 from the variable motion of the moon ; and to refer to the observer the 

 constant which involves the various forms of personal equation. Yet, 

 according to the common acceptation of the theory, this form of 

 the equation is allowable, since all the unknown quantities have 

 known coefficients. 



Again, as soon as the equation involves unknown quantities which 

 pertain to different instruments, it becomes so complex in its character 

 that we can no longer assume that even the sum of the terms which 

 affect AT is the most probable value that can be found, for in so 

 doing we assume that AT is a constant, whereas the solution requires 

 it to be a variable. 



Let us now inquire how far these views are confirmed by the facts 

 of observation. 



In my own case, the probable error of a single reading of four 

 microscopes of the meridian circle is ±-094". If, therefore, as many 

 as 10 observations are obtained, the probable error of the mean will 

 be not far from ±.03". The probable error of a single difference 

 between myself and my assistant, Mr. Joseph F. MacCormick, is for 

 a single reading of four microscopes 4-.125". 



The probable error of a single complete observation in declination 

 is, in my own case, about ±-36", and of the mean of 10 observations 

 is 4-.ll/ 7 . The probable error of a single complete observation in 

 right ascension is, for an equational star, ±.026 9- and for the mean of 

 10 observations ±.008 8 -. 



