RELATION OF VARIABLES. 101 



independent variables y, z, etc., are reduced (see p. 128), the expres- 

 sion for w takes the form 



^o=fli^■)+^(:y)+f^{z)+.... (1) 



where/1,/2,/3, etc., denote the unknown functional relations of w to x, 

 y, z, etc. The problem is to determine each of these unknown func- 

 tions from the numerical data. 



In laboratory experiments it is usually possible so to control the 

 independent variables as to hold all but one, say x, at constant values, 

 y, z, etc. In such instances the difference between any two values 

 of the series /i (xi),/i (xo), /i (X3), etc., can be readily found, where 

 Xi, X2, X3, etc., are averages of x in each of a series of groups formed in 

 succession from the values of x arranged in ascending order of magni- 

 tude. The purpose of taking averages is to eliminate so far as possible 

 effects of accidental variations due to variables beyond i.-ontrol. The 

 corresponding averages, Wi, Wo, W3, etc., of the observed values of lo, 

 therefore are 



wi = /i(xi) + M 

 Wo = /i(xo) + M 

 W3 = /i(x3) + M (2) 



where 



w„=/i(xj + il/ 

 M=My)i-Mz) + .... (3) 



is a constant since y, z, etc., are constant. Similarly, the relation of 

 w to y,w to z, etc., may be thus determined. 



But it is only under the artificial conditions of the laboratory that 

 this simple way of determining the unknown functicns is valid; 

 and, even so, there is no guarantee that the same functic nal relations 

 will hold good under natural conditions. In nature one is limited to 

 observing what is actually taking place; all influences are beyond 

 control; all vary simultaneously; and all are more or less correlated. 

 Differences between successive values of w (equations 2) are therefore 

 not due, in general, to the fluctuation in x alone, but also to fluctua- 

 tions in the remaining variables, y, z, etc. Moreover, the effect of 

 these remaining variables often is large enough to produce serious 

 errors in the relations indicated by this simple mode of procedure. 



Such errors must be eliminated. To accomplish this, say in the 

 relation of lo to x, corrections are computed for the purpose of reducing 



