4o6 



SCIENCE PROGRESS 



If P comes out high, e.g. -So, then our formula is probably the 

 correct one ; if P comes out low, e.g. -05, the formula is a very 

 bad fit. It is not possible to state where a good fit ends and a 

 bad one begins. 



As an illustration of the method, I have made a special set 

 of observations on a glass slab, determining the angle of refrac- 

 tion three times for each of the eight angles of incidence 10°, 

 20°, 30°, 40°, 50°, 60°, 70°, and 80°. The apparatus employed 

 was exceedingly simple, one of the arrangements used in schools 

 under the name of " Pin Optics." The following table shows 

 the calculation : 



MeaQ Sine of 



Angle of 

 Refraction. 



•1059 

 •2184 



•3244 

 •4I7I 

 •4969 



•5634 

 •6020 



•6328 



Theoretical 

 Value. 



•III9 

 •2206 

 •3224 

 •4145 

 •4940 



•5584 

 •6059 



•6351 



Difference. 



+ 

 + 

 + 

 + 



Refractive inde.x 



•0060 

 22 

 20 

 26 

 29 



50 

 39 

 23 



(Difference)i. 



•00003600 

 484 

 400 

 676 

 841 

 2500 

 1521 



529 



Sum •00010551 



Sum of Squares of 



Differences of Observations 



from Mean. 



•00000241 

 6260 

 7690 



728 

 2681 



632 

 2393 

 3358 



Mean ^0000299 



In the table the first column gives the mean of the three 

 observed values of the sine of the angle of refraction, the second 

 the theoretical value, the third the difference of these two 

 quantities, the fourth this difference squared, and the fifth the 

 sum of the squares of the differences of the three observed 

 values from their mean. Then by means of the formula 



r 



^\ n{m — 



ni 



)'\ 



y 



where m is the mean experimental value, m the theoretical 

 value, n the number of observations of sin r to each value of i, 

 and 0-' the value which the square of the standard deviation 

 would take for an infinite number of observations at the value 

 of t in question, a quantity x' is calculated. o-« is the weak 

 point in the calculation, since the value of n is so small. I 



have simply taken a' = ^ (average of last column). For a full 



discussion of the method, reference should be made to the 

 original paper. I find 



, 2 X •0001055 ^ 



X = ■ — = 7*o6, 



-0000299 



