Changes of Curvature by Means of a Flexible Lath. 439 



twice the average experimental error e, m the number of 

 points exhibiting this excessive error, and n the total number 

 of points in the drawing. If e 2 is less than unity, unity 

 should be substituted for it. 



Referring to this error, Mr. Hayes has recently written 

 to me as follows : — " I have been looking into the values 

 for your e 2 error, and find reason to think them perfectly 

 fair, or, if they are unfair at all, it is in giving too modest 

 an estimate of this error. The ideal values of e 2 I find 

 to be about -J, and any value for it greater than, say, 

 2 would be highly improbable. With a limited number 

 of points, the fact that e 2 was less than ^ (or 1) would prove 

 nothing. I am inclined to think something involving the 

 ratio of the found sum, s, to the theoretical sum, s' ', would be 

 more definite. Your numbers are hardly large enough for 

 big values of s, since they vary approximately as s only. I 



should suggest / — ) as a suitable expression for e 2 , calling 



it only 1 if it were less than, say, about 3 or 4." 



3. The probability of a drawing will generally vary in some 

 inverse ratio with the number of errors with like signs which 

 it represents as occurring consecutively. A group of 6 errors 

 with similar signs is evidently less probable than two separate 

 groups in different parts of the drawing with three each, so 

 that we cannot get any estimate of the relative probability of 

 different drawings by comparing the sum of the numbers of 

 errors with like signs occurring in groups ; nor will, I think, 

 the sum of the squares of these numbers lead to a just estimate, 

 for it appears to me that a group of 6 like errors is more than 

 twice as improbable as two groups of 3 each. I have therefore 

 taken the sum of the cubes, and as the actual number thus 

 obtained will be dependent on the total number of experiments 

 dealt with, I have divided this sum by 5 times the number of 

 experiments — 



m 1 3 + m 2 3 + . .. 



e* 



5/i 



In determining m Xl m 2 , &c. the intervention of a nil error is 

 counted as a change of sign ; and, as in the previous case, the 

 value of e 3 is never allowed to fall below unity. It will = 1 

 if, say, in twenty points we get ten groups of 2 similar signs, 

 four groups of 3, or one group of 4 with one group of 3 and 

 one group of 2. 



I was led to select 5 as the coefficient of n by determining 

 what coefficient was required to make e z — \ in the case of 

 numerous series of imaginary errors constructed by drawing- 

 lots for the signs and casting dice for the magnitude. For such 

 a purpose the faces of the dice must be marked so as to corre- 



