282 REPORT— 1887. 



sense might be lightest m the latter sense. Another caution is to distinguish the 

 present investigation from that whose object is the displacement of the centre of 

 (iravify of the system/ a qvcesitum whicli does not presuppose any common dis- 

 turbing agency. 



Again the pi-oblem special to this section has been hkened to the problem of dis- 

 covering the proper motion of the solar system by means of the apparent move- 

 ments of the stars. Let us suppose, for the sake of illustration, that the line in 

 which the solar system moves has been ascertained. The only questions are in 

 which direction of that line, positive or negative, say towards or from a certain 

 star in Hercules, and at what rate, we are moving ; how far we have moved 

 between two given epochs. Now, if we take several groups of stars at random, 

 say (as in fact is done) some groups in the northern hemisphere, and others in the 

 southern, and for each of these groups we take the mean of the apparent motion of 

 the stars along the given line ; then, if the mean resultant is much the same ^ for 

 every group, we may be reasonably certain that the phenomenon is due to a 

 common cause, which is doubtless no other than the proper motion of the solar 

 system. Suppose, however, that the motions of the stars did not conform to what 

 may be called a true mean. Suppose that what Mr. Proctor calls ' star-di'ift ' 

 was prevalent on a much greater scale than he has found to be the case ; that the 

 Milky Way, together with other zones, moved oft'*?; hloc in one direction, while the 

 Great Bear carried off another half of the heavenly host in the opposite direction. 

 In this case we should no longer be able to detect the motion proper to the solar 

 system. The peculiar grip which a plurality of independent events affords to the 

 calculus of probabilities now becomes wanting. 



It is to be observed that, in assigning importance to the different indications 

 given by the apparent motions, the criterion is not the mass of the star, but its 

 ' weight ' ^ in the sense of aftbrding a better measure of the qucesitum, the motion 

 of the solar system. 



Similarly, in the problem before us it must be either given by previous e.xpe- 

 rience (as in the case of our first illustration), or discoverable from the data them- 

 selves (as in our second iUustratiou), that there is a true mean ; that one set of 

 commodities, such as the products of extractive labour, has not risen en hloc, while 

 another set, as manufactures, has fallen. Without that condition we cannot follow 

 Jevons in reasoning bythe principles of probabilities that gold has been depreciated 

 (or appreciated) to a certain extent. With that condition we may follow Jevons 

 in taking a mean of price-variations, irrespective of the quantities of the com- 

 modities. 



The problem before ns may be thus defined. Given a number of obser- 

 vations consisting each of the ratio between the new price and the old price 

 of an article, to find the mean of these observations — the objective or quasi- 

 objective mean — as distinguished from those combinations in the pre- 

 ceding sections which were prescribed by considerations of utility. The 

 problem as thus conceived belongs to that higher branch of the calculus 

 of probabilities which may be called the doctrine of errors. Upon the 

 theory of errors are based two kinds of problem ; of which the first is 

 exemplified by the method of determining the true position of a star from 

 a number of separately erroneous observations, the second, by the method 

 of constructing the typical stature of a people, Vhomme 7noyen, from the 

 measurement of a great number of individuals. To which of these ana- 

 logies — the more, or the less, ' objective ' species of mean — our case most 

 corresponds is a nice inquiry, varying with the shades of hypothesis.* 



' Analogous to the calculation of Units in our earlier unhypothetical sections. 



- If it be asked what extent of difference between the means of different groups 

 is to be expected and may be regarded as insignificant, the answer is supplied by the 

 mathematical Theory of Errors. See the writer's paper on Methods of Statistics. 



^ Depending on considerations not here relevant. 



■• Consider the illustrations given below at p. 293 



