282 MEMORIES OF MY LIFE 



analogous problem that interested me a few years 

 previously [158]. I have had more than once to 

 assist in determining how a given sum allotted for 

 prizes ought to be divided between the first and 

 second men when only two prizes are given. The 

 same problem has to be solved by the judges of 

 cattle shows, and it is, if a little generalised, of very 

 wide application. I attacked it both theoretically 

 and practically, and got the same results both ways. 

 When the number of candidates is known, and the 

 distribution of merit follows the well-known Gaussian 

 law, the calculation is easy enough, but when the 

 number of candidates is not known it is a different 

 matter ; moreover, the Gaussian law may not apply to 

 the case, though it will probably do so pretty closely. 

 So I calculated what the ratios would be in classes of 

 different numbers and according to the Gaussian 

 law. The ratio in question is that between the 

 excess of the first performance over the third, and 

 the excess of the second performance over the third. 

 The third being the highest that gets no prize at all, 

 forms the starting-point of the calculation. When 

 the numbers of candidates were either 3, 5, 10, 20, 50, 

 100, 1,000, 10,000, or 100,000, I found, to my surprise, 

 that the ratio was much the same. The appro- 

 priate portion of the total of one hundred pounds 

 which should be allotted to the first prize proved to be 

 seventy-five pounds, leaving twenty-five or one-third 

 of its amount for the second prize. Even when the 

 number of candidates were at the minimum of 3, the 

 first prize would be 6j ; if 5, it would be ji ; if 

 10, it would be ^73; and if 100,000, it would be 

 (to the nearest whole figures). 



