THE LAW OF REFRACTION 405 



have made all possible allowances, there may not be a remainder, 

 a systematic deviation between theory and observation. All 

 that we can say at present is, that there is no trace of such a 

 deviation, that there is nothing whatever to suggest that the 

 law, as we have it now, is not absolutely true. It would be 

 an almost impossible problem to calculate the probability that 

 it is absolutely true, since it has been verified so repeatedly and 

 in so many different ways. But when we deal with an isolated 

 series of observations we are on surer ground. Prof. Karl 

 Pearson states that he looks forward to the time when no 

 physical paper will contain a curve fitting a series of observations 

 without some estimate of the probability that the law is abso- 

 lutely true, without some estimate of the goodness of fit, i.e. of 

 the percentage of trials in which we should get in random 

 sampling a fit as bad or worse. He has provided the necessary 

 formulae * and tables by which the estimate can be made. 



Prof. Karl Pearson's methods are as yet quite unknown to 

 physicists, parti}'" because there is no good textbook explaining 

 them, and partly because statistics is not yet an " examination 

 subject." But on account of their great importance it is 

 desirable to illustrate their bearing on the law of refraction here. 

 Considerations of space make it impossible to go into them fully. 



Suppose that for a series of angles of incidence we determine 

 the corresponding angles of refraction with the utmost refine- 

 ment that modern instruments are capable of, making several 

 determinations of the angle of refraction for each angle of 

 incidence, then there will be differences between each mean 

 experimental result and the corresponding theoretical value. 

 Differences are naturally to be expected owing to human 

 fallibility and imperfections in the instrument, owing, for 

 example, to looking at the vernier obliquely, not setting the 

 cross-wire exactly on the middle of the image, or to errors in 

 the graduation of the scale. The question arises as to whether 

 the differences observed are satisfactorily accounted for by such 

 accidental variations, or whether they indicate a real deviation 

 from the law. The physicist usually answers this question in 

 a rough manner by drawing in the theoretical curve, plotting 

 the observed values, and seeing how close they lie to the curve. 

 It is possible to be misled by the scale of the curve. Prof. 

 Pearson answers it more accurately by stating the probability 

 or percentage of cases in which, if the theoretical value were 

 absolutely true, the agreement would be as bad or worse. For 

 example, if the probability P comes out -60, then, if we repeated 

 the experiment 100 times under the same conditions, in 60 per 

 cent, of these cases the agreement would be as bad or worse. 



1 Biometrika, p. 239, 11, 1917. 



