a priori Probabilities. 205 



will have to be modified by adding to the left-hand member 



the term *V { , Again, in the problem* which may be placed 



next to the above in point of simplicity, viz. given a set of 

 observations diverging from a known point according to a 

 sought modulus, it has been tacitly assumed that, a priori, one 

 value of the modulus is as likely as another. But, if it is not 

 so, let %(/*) A/i express the distribution of a priori probability. 

 Then the equation to zero, from which the reciprocal of the 

 modulus squared is found to be half the reciprocal of the 

 mean square of error, will have to be modified by adding to 

 the left-hand member the term 



n-l 



A fortiori in the problems compounded of those simple cases ; 

 in particular those which we distinguished in a former paperf 

 as Probs. III. and IV. And if we ascend to the higher pro- 

 blemsj, which are not restricted to exponential laws of error, 

 and to the more sublime qusesitum of utility as distinguished 

 from probability, there also does the necessity of an a priori 

 foundation confront us. A priori also is the basis of the im- 

 portant theorem which determines the probability that two 

 sets of observations have diverged from different means. 



(2) The preceding remarks relate primarily to the measure- 

 ment of objective continuous quantities, real spaces and times 

 — for instance, a star's position in the heavenly sphere, or its 

 time of crossing the meridian. What is true of this class of 

 measurables is true, mutatis mutandis, in the case of integer 

 numbers, and of the Means which have been called subjective, 

 fictitious, or typical. An example of a real integer number 

 estimated by a sort of method of observation, is Jevons's§ 

 calculation of the number of sovereigns in currency. Ex- 

 amples of the fictitious Mean are : — in space an average group 

 of barometrical heights, in time the average flowering-time of 

 a plant, in integer number all the Kegistrar-GeneraPs returns. 

 We are not now concerned to examine the distinctions which 

 have just been indicated. For the present purpose we may 

 compare with the first problem of the preceding paragraph 

 the statistical practice of taking the mean of a large number 

 of figures. The analogues in integer number of the problems 

 in continuous quantity which involve the determination of a 

 modulus are less familiar. Statistical examples resembling 



* Phil. Mag. 1883, vol. xvi. p. 366. f Ibid. 



t PHI. Mag. February 1884. 



§ ' Investigations in Currency and Finance,' Essay IX 



