( wi; ) 



never be the same and also differ according to the combination that 

 one has in view. As, however, we have never mutually combined three, 

 but always two odours at a time, it will never be necessary to occupy 

 ourselves with the units of length of the three at the same time, nor 

 does this change of units, depending on the case considered, raise any 

 objection. Even in our further demonstration this does not give rise 

 to any difficulty, as we are never going to mutually compare vectors 

 but when they have the same direction with regard to an independent 

 vector that is at the same time considered, in other words possess 

 with respect to the latter about the same units of length, 1 ). 



A third set of vectors can, speaking generally, not be placed in 

 the same system, even though it has one vector in common with 

 the two preceding systems, for the fourth vector will in general 

 have to be given different directions, according as it is considered 

 in connection with the first and second, with the first and third or 

 with the second and third. But what is in general impossible, may 

 in special cases prove quite practicable. Let us consider this. 



If we number our nine standard-odours with the figures 1 to 9 

 and likewise the corresponding vectors, each lime two of these 

 vectors can be fixed and the rest arranged with regard to these two 

 vectors, which are definite in their situation. The question we put 

 just now, comes to this: Is the mutual relation between the odours 

 perhaps so as to make some of (hese last seven vectors coincide ? 

 In consequence of mistakes in the experiment a complete coincidence 

 will no doubt be out of the question, but let us consider whether it 

 happens within a margin of* error of at most 1 % 0l 2ar difference 

 of direction (= 3.6 C ). For this purpose we have first combined 1 

 and 2, considering all the others with regard to these two; then 1 

 and 3 are fixed, the rest arranged according to this, etc. till all 

 combinations, 36, have occurred. In each of the combinations seven 

 vectors were met with, whose situation with regard to the two 

 vectors previously chosen had to be traced in order to see whether they 

 coincided or not. For each set of two previously determined vectors 

 this gives rise to 42 judgments, so that in all 1512 judgments have 



!) The proportional numbers as they have been empirically composed and taken 

 together in our table, form 252 possible constellations of three vectors. Among 

 them there is only one which, also as to the units, is completely satisfactory for 

 all three proportions at the same time. It is the constellation in which terpineol, 

 scatol and valerian acid are combined. The length of the vectors measured by 

 means of a joint unit of length amounts in this case to I for the terpineol-vector, 8 

 for the scatol vector and 20 units for the valerian acid vector. 



