70 MATHEMATICAL BIOPHYSICS OF THE CENTRAL NERVOUS SYSTEM 



•7 



.6 - 



.1 



Tactile data: Macdonald and Robertson 



o <33< 



1.5 



2.0 



2.5 



log! 



Figure 8. — Comparison of theory with experiment: intensity-discrimination 

 at varying intensity-levels for visual, auditory, and tactile sensations. Solid 

 curves, theoretical predictions by equations (25) -(28) ; points and dotted curves 

 experimental. Tactile data from Macdonald and Robertson, 1930. Abscissa, in- 

 tensity (on logarithmic scale) of stimulus; ordinate, ratio of just-discriminable 

 difference of total intensity. 



and evidently u(S + S6) =w(l + 6). Writing equation (26) with 

 S replaced by S + S6 , and introducing (28) we obtain 



ILX" 



(36u + 1) x ? + (S6 2 u + 26) x- (u6 3 - 26 - 1) = . (29) 



By eliminating x between equations (29) and (26), we obtain 6(u), 

 the desired relation between the Weber ratio 6 and the intensity of 

 the stimulus. The result can be expressed in the form (Householder, 

 1942c) 



S = pu- a ^ (30) 



where a is very nearly constant and lies between 1/2 and 1/3 . 



In Figures 6 to 8 are shown a comparison between theory and ex- 

 periment for visual, auditory, and tactile data (Householder, 1939). 

 The abscissa log u is used for convenience; u is proportional to the 

 stimulus S . It should be emphasized that there is but the one para- 

 meter involved in each curve. 



In Figure 9 is shown the theoretical and experimental results for 

 the case of visual discrimination of lengths (Householder, 1940). In 

 order to make the comparison it is necessary to assume only a propor- 

 tionality between the length of the line and the value of e resulting 



