a priori Probabilities. 209 



not equally probable. The hypothesis is thus shown to be 

 inconsistent, since it cannot be applied to these two unknown 

 events of Y and Y repeated." Of course it must be admitted 

 that, if a certain variable has as often one value as another, a 

 second variable depending upon the first may not enjoy that 

 equal distribution of values. If the stars are distributed at 

 random over the heavenly sphere, then the distance between 

 a star and its nearest neighbour will not have one value as 

 often as another. Very great and small distances will be 

 rare. If cannon-balls be fired with constant velocity from a 

 cannon which is directed at random to any point on a sphere 

 described about the turning-point as centre, the balls descend- 

 ing will not strew the earth uniformly like snow. And, to 

 come to the point, if the probability p of an occurrence X 

 is uniformly distributed between and 1, then the probability 

 P of the double event x, x will not be distributed uniformly, 

 but according to this law: that the probability of the a priori 

 probability being between P and P + AP is (not AP) but Ap, 



that is in the limit AP -^, that is AP — j-, since P=p 2 . 



Which seems contrary to experience. 



(3) Let it be granted, however, that Boole and his fol- 

 lowers hit a weak point if they suggest that, though some- 

 thing in each class of phenomena occurs as often one way as 

 another, it is not always possible to determine what it is which 

 is thus equally distributed, whether our cannon is aimed indif- 

 ferently in every direction, or our cannon-balls fall uni- 

 formly like snow. Still we may fall back upon the position 

 that, where the form of a function is completely unknown, it 

 is allowable to assume that form which is most convenient for 

 the purpose of calculation (especially where we have reason 

 to suppose that the results of different hypotheses are not 

 widely different). It is upon this principle that Laplace took 

 the first step in his celebrated method of Least Squares, 

 assuming that the qusesitum is an arithmetic mean of the ob- 

 servations. If, indeed, the views put forward in a former 

 paper in this Journal are correct, that step is by no means so 

 precarious as has been supposed. Still that Laplace should 

 have taken that step in the dark and that it should turn out 

 to be correct is very instructive. The same principle must 

 be employed in the determination* of a change in the general 

 purchasing power of gold, and probably in many other parts 

 of Social Science. The principle seems especially appropriate 

 to cases where we seek, not so much an exact measurement, as 



* ' Statistical Journal/ Jan. 1884: 

 Phil. Mag. S. 5. Vol. 18. No. 112, Sept. 1884. P 



