290 REPOKT— 1887. 



If the objects casting shadows were unsteady — waving trees, for instance 

 — a single measurement might be insufficient. We might have to take the 

 mean of several shadows. Now for our purpose the breadth of the upright 

 object casting the shadow would be unimportant. The ' wide-spreading 

 beech ' and the mast-like pine would serve equally well as a rude 

 chronometer. 



Suppose, however, that the top of the broader tree was not level but 

 serrated, each apex oscillating more or less independently. If by the 

 shadow of a tree was understood the mean length of the shadows cast by 

 all its apices, in that case the broad tree should count for more than a 

 bare pole. How much more, would depend upon the connection between 

 the projecting branches. The more independent the oscillations of each 

 apex, the better the measure aflPorded by their mean shadow. 



This image seems appropriate to our problem. Each price which enters 

 into our formula is to be regarded as the mean of sevei-al prices, which 

 vary with the differences of time, of place, and of quality ; by the mere 

 friction of the market, and, in the case of ' declared values,' through errors 

 of estimation, it is reasonable to suppose that this heterogeneity is 

 greater, the larger the volume of transactions. On this account, therefore, 

 and irrespective of those considerations of utility which were proper to 

 our earlier sections, greater weight should attach to the prices of those 

 commodities whose quantities are larger. It does not follow that the 

 weights should be proportionate to the masses. The proper coefficients 

 could be ascertained by scientifically examining the detailed statistics of 

 each market. But it is agreeable to the Theory of Errors ' and to the 

 successful practice of physicists to employ a discretionary good sense in 

 assigning ' weights ' when a precise determination is difficult or impossible. 

 In our case a good system of weights appears to be affijrded by the 

 quantities of commodities sold (once, and exclusive of resales) per unit of 

 time. The weight so assigned would doubtless often be too large. It 

 might sometimes be too small in the case of commodities much resold. 

 On the whole it would be a good and safe system. This principle of 

 ponderation is to be combined with those which have been given in the 

 last section.- If we suppose the variation of prices not confined to a par- 

 ticular zone, but propagated over the whole sphere of industry, then we 

 shall obtain a set of weights almost coincident with those prescribed (upon 

 a different ground) by the standard based on National Consumption 

 (Section III.). For the condition that the observations should be inde- 

 pendent,^ leads us to exclude, or at least take little account of, the same 

 commodity at different stages of production.'* 



' An improvement in weighting can only diminish, very often only slightly 

 diminish, the error inevitably incident to the result of any measurement. 



- See the headings o, fi, y, p. 287. 



' See /8, loo. cit. 



* It would be a question whether industrial wages and industrial rent should be 

 included, in addition to, and otherwise than representative of, the corresponding 

 products. At any rate their weights ought not to be proportionate to their volumes ; 

 partly on account of their close connection with commodities, partly on account of 

 the magnitude of these volumes. In the case of transactions so extensive, and 

 perhaps we may add some other large interests such as cotton and iron, it would be 

 best to determine the proper coefficients by specially examining the detailed statistics 

 of each market in the light of the Theory of Errors. A summary method would be 

 to assign to these enormous masses an averagely large weight about as large as any 

 other weight employed in our operation. The ideally best weight is not likely to be 

 very different from the arbitrarily assigned one, and slight differences of weight do 



