COMPARISON OF CATALOGUES OF STARS. 397 



Each of these methods has its strenuous advocates. The objection most frequently 

 urged against the first method is that in an analytical solution, extended to terms of 

 the second order only, there can be only two points of inflexion in a curve represent- 

 ing the computed residuals. If therefore the residuals between two systems are really 

 represented by a curve which has several points of inflexion, the analytical method 

 must be less exact than the method of graphic curves. Again, it often happens that 

 residuals having the same sign and nearly the same magnitude are concentrated at 

 one point. The effect of the analytical solution is to distribute the assumed errors at 

 this point over the whole system in such a way that the sum of the squares of the 

 residuals shall be as small as possible. But if these errors are real, they belong to 

 this point only, and they should not be distributed over the whole system. 



The objection to the second method is that for a given series of points representing 

 Aa or AS, no two persons would describe exactly the same curve. This objection is to 

 a certain extent obviated by the plan of successive bisections of the lines joining the 

 points, described by Professor Pickering in Vol. X. of the Annals of the Observatory. 

 In drawing a curve b} r this method, two points require attention. First, that the 

 uncertainty with regard to the ends of the curve will be increased in proportion to 

 the number of bisections, and second, that each successive bisection causes the curve 

 to approach a straight line. In order to draw a regular curve, it will rarely be neces- 

 sary to make more than three bisections. It is my custom to use the curve drawn 

 through the final bisecting points as a guide in drawing the curve most nearly repre- 

 senting the observations. To this extent, therefore, the final curve is dependent upon 

 the judgment of the computer. The uncertainty thus introduced will rarely exceed 

 .005" for any residual in a, and will, I think, never exceed .01 s . It is obvious that the 

 case in which this uncertainty is the greatest is that in which there is a sudden change 

 in the direction of the curve, followed by a nearly flat curve for three or four points 

 of the horizontal argument, and this in turn followed by a sharp return to the original 

 direction. 



As illustrations of the degree of conformity which may be expected from the two 

 methods, I give below in parallel columns the observed residuals, the residuals derived 

 from the equations, and those derived from the graphic curves, for the following 

 authorities : — 



Catalogues co?ri])ai'ed. Equations. 



I. Auwers minus P' ; A« from +70° to +85° A« = +.069 +.065 m +.041 n +.029 m' +.011 n> 



II. Auwers minus H..C. Or, A« from +70° to +85° A« = +.033 +.007 m +.029n -.009m' -.001n' 



III. Auwers minus Struve; AS from +70° to +85° AS = +.03 -.08m — .20n +.03m' +Mn' 



IV. Auwers mjMMsSaffbrd; AS from +10° to +70° AS = -.09 +.57 e -.48 d 



VOL. x. 51 



