Adjustment of Observations. 185 



In accordance with the usual terminology, we may call 

 y/dX 2 the error of a single mean X, and p b , p CJ . . . . p )H 

 the weights of 6, c, . . . m. For a given accuracy of the 

 observations, the calculated values of b, c, . . . m will be 

 the more accurate the greater the weight. 



If there are only two magnitudes a?, y and only two 

 constants &, m we have 



(Yi-Y 2 ) 2 (^-Y,) 8 



/>& = 9 ; ffm= -y 2_y 2 ' ' * ^ ' 



If there are three magnitudes and three constants b, c, m 

 we have 



(Y,Z, -Z 1 Y 2 + Y 8 Z 3 - Z 2 Y, + Y 3 Z 1 -Z,Y 1 ) S 

 ^- (Z 1 -Z I )« + (Z»-Z# + (Z 1 -iy» 



_ (Y^— 2^2 -fl^Za — Z 2 Y 3 4- Y 3 Z 1 — ZsIlj) 2 ._> 



P '~ (Y.-YO'+tY.-Y^ + C^-Y,)' ; 



(YiZi-Z^ + Y^-ZsYa+ Y sZj-Z^!) 2 

 '" ~ (-YA-ZxY,)' + (Y 2 Z 3 - Z 2 Y S ) 2 + ( Y^Z, - Z S Y,) 2 J 



From (6) we reach again the conclusion that the calculated 

 values will be made most accurate by making Y 1 and Y 2 as 

 different as possible, which is effected by grouping the obser- 

 vations in the manner proposed. In (7) the matter is more 

 complicated, and the most accurate way of grouping the 

 observations depends on the values of b, c; but in many 

 important cases the rule which has been proposed gives 

 the calculated values the greatest possible weight, and in 

 no case does it seem to give them a weight very much less 

 than the greatest possible. When ease of application is 

 taken into account it is improbable that any more suitable 

 rule of general validity could be found. 



By the aid of (4) and (5) probable errors of the calculated 

 values can be found in a manner exactly similar to that of 

 the method of L.S. We shall inquire later what is the 

 significance of such probable errors according to L.S. or Z.S., 

 but for the present we shall use them merely as a rough 

 method of comparing the results obtained by the two methods. 

 It should be observed that when (4) is used in L.S. to obtain 

 the probable error, dx 2 occurs in place of dX 2 , where dx 2 is 

 the mean square error of a single observation and dX 2 the 

 mean square error of the arithmetic mean of p such single 

 observations. In our estimates of probable error by Z.S., 



