ON ERRORS OP OBSERVATION. 157 



denominator 24 and | i s not -586, but '583. The 

 results nearly coincide, but so do the data, for which 

 reason the most probable result may be very nearly 

 found, or of two results which differ very little, one 

 may be much more probable than another. When 

 observations give magnitudes so nearly coinciding as '6 

 591 and '567, it is worth while to examine the relative 

 probabilities of methods which give results so nearly 

 equivalent as '586 and '583. Which of the two 

 preceding methods is most entitled to confidence ? 

 Analysis points out that this question is useless, because 

 there is a third method which is more safe than any 

 other. The method of least squares in the case before 

 us leads to the following rule ; W r hen both the 

 numerator and denominator of a fraction are to be 

 determined by observation, and various corresponding 

 observations of both are made, multiply each numerator 

 and denominator by the denominator, and divide the 

 sum of the numerators so formed by the sum of the 

 denominators. Thus in the preceding instance, it is 



12x20 + 13x22 + 17x30 1036 



or '581 



20 x 20 + 22 x 22 + 30 x 30 1 784 



which is more probable than either '586 or '583. 



If the mean risks of all the observations be the same., 

 the mean risk of the preceding result is found by adding 

 1 to the square of the result obtained ('58 1) dividing 

 by the denominator which produced it (1784), extract- 

 ing the square root of the quotient, and multiplying 

 the mean risk of each observation by this square. 

 Thus -581 X -581 is -337561, which divided by 1784 

 gives -000189, the square root of which is -014. The 

 mean risk of the result, therefore, is less than one 

 seventieth part of that of each of the observations. 



The method of least squares is an extension of that 

 of taking an average, or rather it indicates the most 

 probable average in cases which, by reason of more 

 results of observation than one being involved, an in- 

 finite number of different averages exists. It is not 



