FROM NUMBER TO QUATERNION. 75 



the result must be such that added to b^ it gives a o 



But a + b fulfils that condition for 

 6^ + 0^ + 60 = a + (b + &„.) (commutativity and associativity of 



addition) 

 and as b 4- b^ = o (definition of addition) 

 therefore 6 7r + a + 6 =a 

 so that a Q — b^ = a -f & 



that is to say the subtraction of b^ comes to the same thing as the 

 addition of b 



In a similar manner we could prove that 



a o — 



b o = a o + K 



a 7T — 



K = a ir + b o 



a ir — 



h o = a 7T + K 



so that we can say in a general manner that the subtraction of a 

 directive quantity is the same as the addition of the directive 

 quantity having same tensor and different argument. 



Moreover the subtraction is now an operation having always'a 

 meaning. 



Multiplication. — To multiply two directive quantities is to make 

 the product of their tensors and the sum of their arguments or in 

 other words : 



We will call the product of two or more directive quantities, 

 that quantity having as tensor the product of the tensors, and as 

 argument the sum of the arguments of the quantities considered. 



That is to say by definition 



a .b = {ab) 

 a w .b = (ab) v 

 a o • K = ( a & )tt 

 a *K = M) 2 7r = ( ab )o 

 The first equality shows that our definition includes as a 

 particular case the muliplication of absolute quantities, therefore, 

 without any further consideration our definition is a correct one 

 as satisfying the only condition it must satisfy. 



