MULTIDIMENSIONAL PSYCHOPHYSICAL ANALYSIS 77 



these cases to which the present method does not apply, we suppose, 

 therefore, that for some set of these objects, say A (0) , A (x) , and A (2) , 

 the S of each pair is exceeded by the sum of the other two, and we 

 attempt a two-dimensional representation. It is evident that the val- 

 ues of the various S's alone can determine the various A» at most up 

 to an additive constant — the zero for each modality is arbitrary un- 

 less prescribed by considerations irrelevant to the formal experi- 

 mental procedure. Such prescription, if available, may be observed 

 later by an appropriate adjustment; for the present choose A <0) as 

 one reference-point and assume that each Ai (0) = . If S {pq) repre- 

 sents the S corresponding- to the pair (A (p) , A (9) )» we may suppose, 

 further, that 



£(02) > gCL2> 



relabeling the objects if necessary. For determining the four quan- 

 tities A (i) j, only three equations are available. Hence we may make 

 the assumption 



r±l -f±2 ) 



subject to possible later revision. Finally, we may suppose that 

 A x {2) < A 2 (2) , since we can only separate but not identify the two 

 modes, and we have 



A x < 2 > = [£<° 2 > - £< 12 >]/2 , A 2 < 2 > = [S< 02) + S< 12 >]/2 , 

 A^ = A^ = S (01 > /2 . 



Now consider any object A (3) . If either 



£(03) =£(01) _|_ £(13) —£(02) _|_ £(23) 



or else 



£(03) = £(01) _ £(13) — £(02) _ £(23) 



then 



£(03)— ^(3) + 4 2 <3) 



and the quantities on the right are indeterminate. If neither, there 

 are at least two independent equations involving A a (3) and A 2 (3) . If 

 there are three, a two-dimensional representation is impossible, but 

 if for every fourth object A (3) there are at most two new equations, 

 two dimensions are sufficient. 



Thus, apart from the arbitrariness indicated, with a sufficiently 

 large number of stimulus-objects A all the Ai can be determined. It 

 is perhaps clear enough from the above how one must proceed when 

 more than two dimensions are required. The quantities S may be 

 regarded as distances in the representative space, and the space is 

 metric but not Euclidean. The assumption of linearity imposed upon 

 the mechanism is not highly restrictive, in principle, since, if the 



