what is commonly called Fechner's Law, which he has established by aid of an enormous amount of careful experiment in the case of a number of our simple sensations. But I do not believe that he has made any claim that such a law holds good in the far more intricate dependence of happiness upon wealth.
10 The formula expressive of this moral happiness is c log x/a; where x stands for the physical fortune possessed at the time, and a for that small value of it at which happiness is supposed to disappear: c being an arbitrary constant. Let two persons, whose fortune is x, risk y on an even bet. Then the balance, as regards happiness, must be drawn between
or log x2 and log(x + y)(x − y),
or x2 and x2 − y2, the former of which is necessarily the greater.
11 This may be seen more clearly as follows. Suppose two pair of gamblers, each pair consisting of men possessing £50 and £30 respectively. Now if we suppose the richer man to win in one case and the poorer in the other these two results will be a fair representation of the average; for there are only two alternatives and these will be equally frequent in the long run. It is obvious that we have had two fortunes of £50 and two of £30 converted into one of £20, two of £40, and one of £60. And this is clearly an increase of inequality.