400 PROCEEDINGS OF THE AMERICAN ACADEMY 



corrections required. The general tendency of modern practice is 

 towards the graphic method, in preference to a rigid analytical de- 

 termination. But the difficulty exists that computers do not agree in 

 the details of the graphical methods employed, and hence with the 

 same data different results are obtained. It is the common practice 

 to draw the curves which represent the residuals in right ascension 

 with this function as the horizontal argument, but there are several 

 catalo<fues in which the residuals /ia are functions of both the right 

 ascension and of the declination. 



In the choice of any method, it is obvious that preference must be 

 given to that in which the residuals Ja and z/5 are reduced to a 

 minimum, whatever the order or the limits of the groups into which 

 they may he divided. It is believed that this will be best accomplished 

 by the use of what, for the lack of a better term, may be described as 

 double-argument curves. 



Suppose, for example, that all the residuals in right ascension for 

 a given catalogue which fall within the limits of the groups in declina- 

 tion — 10°...+0°, +0°...-flO°, -flO°...-f 15°, etc., are arranged 

 in groups in the order of right ascension. Since the values of the 

 residuals may be assumed to be true (nearly) for the mean of the 

 groups into which they are divided, whether the argument be the 

 richt ascension or the declination, it is obvious that we can choose 

 either the values which are functions of the right ascension, or those 

 which depend on the declination, at pleasure, in the construction of the 

 graphic curves. But whichever is chosen, the residuals derived from 

 the constructed curves no longer represent the mean values which 

 correspond to the other argument; that is, the values which are 

 functions of one argument are derived from the same curve, while 

 those which are functions of the other argument are derived from 

 different curves. 



The character of these double-argument functions are illustrated in 

 the following (fictitious) example. 



