170 PROCEEDINGS OP THE AMERICAN ACADEMY 



values of r and r , and what conclusions can be drawn therefrom con- 

 cerning the precision of x. 



First, it will be seen that the two formulas do not give pre- 

 cisely the same results, and the relation between the results is changed 

 by rejecting one apparently discordant observation. The difference 

 is, however, insignificant when compared with the actual error of 

 observation. In general, the agreement will be more perfect the 

 greater the number of values of x. 



Second, it is obvious that if for x we write, a;±a constant, the 

 values of v will not be thereby changed ; hence the values of r and r 

 will give no indication whatever with reference to the existence of any 

 constant error involved in the values of x. 



Admitting, then, that there is no constant error in the given series, 

 what degree of precision can be assigned to any single value of x, and 

 to the mean value 61".75 ? 



It would hardly seem necessary to call attention to the erroneous 

 assumption that, since the value of r is ± .57", therefore no single 

 value can be greater than 62".32, nor less than 61".18; or that since 

 r is ± .14", therefore the value 61".75 is true within this limit. The 

 refutation of the first assumption is made sufficiently easy by an ex- 

 amination of the separate values of x, but it is not quite so easy to 

 show the fallacy of the second. 



Notwithstanding the absurdity of attempting to assign to the arith- 

 metical mean the degree of precision indicated by the value of r , 

 observers of limited experiences are continually found doing this, and 

 the writer recalls two instances in which professional astronomers have 

 committed themselves to the same fallacy. 



In general, it is entirely unsafe to draw conclusions with respect to 

 the degree of precision to be attached to the arithmetical mean from 

 the magnitude of the probable error, until the signification of the 

 values from which it is derived is defined. 



If the values of x are found by successive readings of the four 

 microscopes of a meridian circle for the same position of the telescope, 

 the sejiarate values are simple functions of the quantity required, and 

 involve only the accidental errors of the observer, either in making 

 the bisections of the divisions of the circle, or in reading the index 

 of the micrometer screws. In this case the probable error of the 

 mean is a tolerably accurate indication of the degree of precision 

 which may be attached to it. 



But the values of x given, represent the observed index errors of 

 the meridian circle of Harvard College Observatory, as derived from 



