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mutually increasing and decreasing in due proportion, the cardinal 

 values of stimulus and sensation. By analogy we miglit speak here 

 of cardinal values of the stimuli counterbalancing each other, leaving 

 altogether out of the question whether this proportion will prove as 

 easily explainable as that which Fkchner has in view. The zone for 

 which the proportional number of the table holds good, may therefore 

 be called the zone of cardinal proportions. 



From the fact that at a simultaneous impression two odours ran 

 neutralize each other, it follows that the action of these stimuli on 

 the organ may be renresented by two vectors, standing as it were 

 for two foi-ces, which in general act more or less in opposite direc- 

 tions, the direction of the vector of the strongest odour (answering 

 to q in the table) being chosen in such a way that the co-sine 

 of the angle that it forms with the continuation of the vector of the 

 weakest odour (answering to p in the table) is exactly equal to 

 the proportion found for p/q in the combination concerned. For in 

 this case the vector of the strongest odour may be thought to be 

 replaced by the sum of two other vectors: one in a direction opposite 

 to the vector of the weakest odour, and one at right angles to it (in 

 the plane of the original vectors). If, moreover, the two original vectors 

 are given equal length, each with such a unit of length as the propor- 

 tional number implies, i.e. for the vector q and its components of q p 

 times more weight than for the vector^, the neutralization of actions 

 that has to be symbolized by the original vectors, will have been 

 accurately expressed. For the vector p and one of the components 

 of vector q will represent equal, but opposite forces. We shall only 

 have to consider the direction of the other component of vector q 

 as direction of odourlessness, in order to have duly accounted for 

 i he complete lack of sensation. 



A second set of vectors can be placed in the same system, pro- 

 vided the two sets have one vector in common. Starting from of a 

 new proportional number p'/q' the new third odourvector that has 

 been introduced, may then be given a definite direction with regard 

 to the first odourvector ; also the second and third vectors may be 

 given their relative directions by means of a third proportional 

 number p"/q"- The latter, it is true, can be done in two ways, 

 according as the third vector is reached by a right- or a left-handed 

 rotation starting from the vector answering to p, but of these two one 

 may be chosen. To the combination p'/q' belongs a vector of odour- 

 lessness at right angles to the vector of weakest odour and to the 

 combination p"/q" a vector of odourlessness at right angles to the vector 

 of weakest odour. The units of length of these vectors will in general 



