372 Mr. D. McAlister on [Nov. 20, 



3°. The law of facility alone will be represented by the curve 



xyV 7r=hexip. ( — Jfi log 2 x). 



Here the area included between any two ordinates, the curve, and the 

 axis of x, measures the probability of a measure lying between the 

 corresponding abscissas. The area of the curve is bisected at x=l, 



but the maximum ordinate is at »=exp. ^— -^pj ( n g- 4). 



Fig. 4. 



(xy\/ 7r = h exp. ( — li 2 log 2 x) . 



I AE = J-exp. (±-\ 



7. (Quetelet's method.) Suppose the measures we are considering 

 to be a series of estimates of the magnitude of some single object 

 (such as the depth of a given tint). In forming such an estimate the 

 mind is acted on by many small causes tending to make us err either 

 in excess or defect. Suppose each small cause acts in such a way as 

 to make us under- or over-estimate in a fixed ratio ; and as a particular 

 case let this fixed ratio be the same for all the small causes. If, then, 

 half the causes tend to over- and half to under-estimation, we 

 estimate rightly. Our estimate in any given case depends on the par- 

 ticular combination of the causes which has (so far as our knowledge 

 is concerned) fortuitously arisen. Let, then, the number of causes be 

 2n, the fixed ratio a. The total number of possible combinations will 

 be 2 2il . The resulting possible ratios are 



„2» „2n-2 „2 ~\ ,,-2 „-2jz+2 „~2n 



at, , as j i . . a j i) a , ... as , as . 



The chance of a correct estimate (ratio 1) is by the theory of com- 

 binations |2^-f-2 2,i \n ]n : the chance of an estimate bearing a ratio 

 aX r to the truth will be |2%-=-2' i \n— r |w+r. Let now n become 

 very large, and a. very near to unity; while r remains finite. Then, 



