458 THE PRINCIPLES OF SCIENCE. 



weight in our mean conclusions, and should bear in mind 

 the discrepancy as one demanding attention. To neglect j 

 a divergent result is to neglect the possible clue to a great 

 discovery. 



Method of Least Squares. 



When two or more unknown quantities are so involved 

 that they cannot be separately determined by the single 

 Method of Means, we can yet obtain their most probable 

 amounts by the Method of Least Squares, without more 

 difficulty than arises from the length of the arithmetical 

 computations. If the result of each observation gives an 

 equation between two unknown quantities of the form 



ax + ~by c 



then, if the observations were free from error, we should 

 only need two observations giving two equations ; but, 

 for the attainment of greater accuracy, we may take a 

 series of observations, and then reduce the equations so 

 as to give only a pair with average coefficients. This re- 

 duction, is effected by, firstly, multiplying the coefficients 

 of each equation by the first coefficient, and adding to- 

 gether all the similar coefficients thus resulting for the 

 coefficients of a new equation ; and secondly, by repeating 

 this process, and multiplying the coefficients of each equa- 

 tion by the coefficient of the second term. Thus meaning 

 by (sum of a 2 ) the sum of all quantities of the same kind, 

 and having the same place in the equations as a 2 , we 

 may briefly describe the two resulting mean equations 

 as follows : 



(sum of cr) . x -f (sum of ab) . y (sum of ac), 

 (sum of ab) . x 4- (sum of b 2 ) . y = (sum of be). 



When there are three or more unknown quantities the 

 process is exactly the same in nature, arid we only need 

 luhlkionnl mean equations to be obtained by multiply- 

 ing by the third, fourth, &c., coefficients. As the numbers 



