FROM NUMBER TO QUATERNION. 77 



Fifth, obviously a a . 1 = a a 



Sixth a a . o = o 



Division. — It is the inverse operation of multiplication and the 

 quotient of two directive quantities is therefore obtained by- 

 making the quotient of the tensors and the difference of the argu- 

 ments, a difference which can always be taken since an argument 

 can always be increased by an even number of it. 



The directive magnitudes we have considered varying in two 

 directions only, are a particular case of more general magnitudes, 

 such as forces and velocities (in a plane), vector radius of a conic, 

 etc., which can take any position whatever in the plane. These 

 magnitudes can be symbolised by straight lines capable of any 

 direction in the plane or by translations and rotations combined. 



Thus AB (Fig. 5) will represt a translation from A to B com- 

 bined with a rotation a from a certain fixed direction OX to the 

 direction AB. 



Extending our notations of the 

 directive quantities previously con- 

 sidered, we can represent AB without 



__* ambiguity by the notation 



O ^ »► -2" 



AB or a n where AB = a 



Fig. 5 a 



AB = a a is called a complex 

 quantity a being the tensor and a the argument. 



It results from our definition that any complex quantity can be 

 transported to any position in its plane, subject only to the con- 

 dition of keeping its direction, that is to say of remaining parallel 

 to itself. And obviously 



a a +2k 7r = a a (k being a whole number) 

 that is to say the argument can be increased or diminished by an 

 even number of it. 



Addition. — To add two complex quantities is to bring the second 

 one in its proper direction to the extremity of the first one, or in 

 other words : 



jr^T... 



