xvn.J THE LAW OF ERROR. 393 



rical and characteristic form of the curve of error. If so, 

 it may be inferred that the errors arise from many minute 

 independent sources, and probably compensate each other 

 in the mean result. Any considerable irregularity will 

 indicate the existence of one-sided or large causes of error, 

 which should be made the subject of investigation. 



Even the most patient and exhaustive investigations 

 will sometimes fail to disclose any reason why some 

 results diverge from others. The question again recurs 

 Are we arbitrarily to exclude them ? The answer should 

 be in the negative as a general rule. The mere fact of 

 divergence ought not to be taken as conclusive against a 

 result, and the exertion of arbitrary choice would open 

 the way to the fatal influence of bias, and what is com- 

 monly known as the " cooking " of figures. It would 

 amount to judging fact by theory instead of theory by fact. 

 The apparently divergent number may prove in time to be 

 the true one. It may be an exception of that valuable 

 kind which upsets our false theories, a real exception, 

 exploding apparent coincidences, and opening a way to a 

 new view of the subject. To establish this position for 

 the divergent fact will require additional research ; but 

 in the meantime we should give it some weight in our 

 mean conclusions, and should bear in mind the discrepancy 

 as one demanding attention. To neglect 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 Simple 

 Method of Means, we can yet obtain their most probable 

 values 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 4- by = c 



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

 need only two observations giving two equations ; but for 

 the attainment of greater accuracy, we may take many ob- 

 servations, and reduce the equations so as to give only a 

 pair with mean coefficients. This reduction is effected by 



