74 G. FLEURI. 



Obviously the argument can be increased or diminished by an 

 even number of ir without altering the quantity under consideration. 



a o = «2fc7T aild a 7T = a (2k + l)7T 



Quantities with argument zero or %kir are absolute quantities so 

 that the directive quantities which we now consider are a generali- 

 zation of absolute quantities or numbers. Our definitions of 

 operations on directive quantities must therefore reproduce as 

 particular cases the definitions already given in the case where 

 the arguments are all equal to zero. 



But these definitions are only subject to that restriction and are 

 otherwise arbitrary * 



Addition. — We will define the operation as follows for two 

 directive quantities AB and CD. (Fig. 4.) 



Let a point move on a straight line 

 OX in the direction AB from to B' 

 so that OB' = AB, then let the point 

 move in the direction CD from B'to D'so 

 that B'D' = CD. 



Fig. 4. 



OD' will represent by definition the 

 sum AB + CD. 



And the definition is readily extended to any number of terms. 

 The fundamental properties of addition resulting from this defini- 

 tion are easily found to be identical with the fundamental 

 properties of the addition of tensors. 



Subtraction. — It is, as already said, the inverse operation of 

 addition, but now those two operations, addition and subtraction, 

 are no more different but become essentially the same operation. 



For let us consider two directive quantities 

 a and b_ 

 and suppose we have to subtract the latter from the former 



* We simply follow the general rule about generalization : As a particu- 

 lar case of the generalized thing we must be able to find the thing we 

 intend to generalize. 



A 





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