WEIGHTS AND MEAN ERRORS. 



One serious difficulty js encountered on the threshold. Whatever principle we adopt for the 

 determination of the relative weight of an observation from the number of comparisons of which 

 it is composed, it is palpable that the accuracy of a position for a given epoch, deduced from 

 successive measurements of the position of a body in motion, is governed by a different law from 

 that which regulates the precision of the mean of numerous independent measurements, with 

 the same implements, of the position of a body at rest. The discrepancies of the several observa- 

 tions from their mean will be distributed according to a somewhat different law in these two 

 cases ; for the circumstances under which the several comparisons are made are constant in the 

 one and slowly varying in the other. The formation of an observation from a number of com- 

 parisons at different times is, in fact, the construction of a normal place ; whether the intervals 

 between the several measurements be counted in minutes, hours, or even days. There can be 

 no reasonable doubt, since the investigations of Professor Peirce upon the theory of errors, that 

 in repeated measurements of the same quantity by the same instrument a limit is soon attained 

 beyond which an increase of precision by an increase of the number of measurements is either 

 absolutely nothing, or, at the most, inappreciable. To determine this limit is a matter of 

 experiment ; and it is very certain that it will not be the same for a variable and for a constant 

 quantity. In fact, the limit for a variable quantity may be regarded as a function of two limits, 

 one of which is the same as for a constant, while the other is not. Therefore our first problem 

 is to determine the relation between the number of comparisons of a planet's limb with a star, 

 and the weight which is to be assigned to the resultant determination of the distance. 



The assumption of a probable error inversely proportional to the square root of the number 

 of observations leads, in fact, to palpable absurdity ; and if rigorously followed would imply 

 that the mean of a very large number of coarse approximations is preferable to that of a few 

 delicate measurements. It may be pardonable to express, in this connection, the strong belief 

 that a practical adoption of some such principle for guidance seems to be exerting a highly 

 prejudicial influence on astronomical observations in many parts of the world ; tending to 

 profligate expenditure of an amount of labor upon the repetition and multiplication of observa- 

 tions, one-fourth of which, if directed to the increase of delicacy rather than the increase of 

 number to quality, in short, rather than to quantity would result in a rapid advance, not 

 only of theoretical and sidereal astronomy, but, through these, of all departments of the science. 

 No multiplication of the number of observations can afford a mean entitled to higher reliance 

 than the nature of the instrument permits, or than the methods, manipulations, and sensibility 

 of the observer are competent to attain ; and the theory of probabilities soon ceases to contribute 

 to the refinement of an accuracy, of which instrument and observer are incapable of taking 

 cognizance. 



Some rule for guidance having been determined upon, we are next to fix the relative weight 

 of positions of the center or measures of diameter, derived from the combination of an unequal 

 number of comparisons for the two limbs. 



The probable error of a pointing, too, is entirely different for a planetary limb, for an 

 estimated center, and for a fixed star ; and finally, the great inequality in the trustworthiness 

 of the places adopted for the comparison-stars exerts its full influence upon the weight to be 

 assigned to the deduced places of the planet. 



These points being disposed of, and values of the several observations of every group being 

 referred to a common unit of weight, we have different groups to be combined with one another, 

 the results of meridian observations with those of equatorial ones, and that uncomfortably 



