274 



EEPOKT 1887. 



articles. In view of the considerations touched in the text the ideally best com- 

 bination of prices must be a complicated fimction, saj' of the form — 0^ »> P ?" -xc-)^ 



F ('PalP'', &c.) 



By an approximation admitted in mathematics, this expression may be written 



ap ^+ p ,, — >_^ -nrhere the weights a, h, &c., are not (like our old friends a, B) 

 (ip^ + hpb + cVC. a , , , . 



quantities, but coefficients deduced from the quantities by the solution of a stu- 

 pendous utilitarian problem. The varying relations between the quantities of 

 things consumed or ' used up ' in manufacture, and the income of different classes 

 — such as the importers and manufacturers in the text (p. 273) — all these complex 

 correlations must be supposed duly expressed by the function F and the derived 

 simpler form. By an allowable abstraction we may suppose the course of industry 

 so uniform that the coefficients a, b, &c., remain constant during the interval under 

 consideration. We shall now show that for the purpose in hand — to mitigate the 

 vicissitudes in each industry — it does not much matter what values (within wide 

 limits) we assign to the weights a, b, &c. As announced in the Synopsis, almost 

 any combination of the more important articles of trade is likely to" be equally im- 

 perfect and equally serviceable. 



Put for ;/., p'/s, &c., the following : p^ (1 + E J, pg (1 + E^), &c. And let the 

 displacements E„, E/g, &c., be made up of two porticjus, one affecting all articles 

 equally, the other proper to each. Call the former e, and let E^ = f + e„, E^ = e + e^, 

 and so on. The unit which would be most desirable in the interest of a single 

 class becomes of the form 1 + e + e^ (putting a single article as the representative 

 of a small group). Meanwhile the general standard is of the form 1 + e 

 Pa^a Pfie/s ^ ^^^ rjijjg j^j.gj. .|. Qf ^Q^jj expressions coincides. But it is onlv 

 ai^a + hpi, 

 by accident that the remainders can be of a piece. For by the theory of errors the 

 displacement (E„) incident to a single article is likely to be of an order much 

 greater than almost any mean of the proper displacements independently incident 

 to n articles. As this proposition turns upon a matter of fact, the independence of 

 the proper displacements of several articles, it may be well to illustrate it by some 

 actual statistics. In the following example afforded by the immense drop of prices 

 during the crisis of 1857, f, the common displacement, is considerable. 



Percentage Deckease of Prices op Several Articles within a Fort- 

 night, November 1857. 

 (Based upon 'Commercial Daily List,' cited by Patterson, Economy of Capital, p. 191). 



Differences 



— Mean square of error 



254 = Modulus sqiiared 



