FROM NUMBER TO QUATERNION. 



79 



Fig. 8. 



Decomposition of complex quantities. — 

 Now draw a perpendicular OY to OX 

 and let OA be any complex quantity a a 

 Forming the rectangle OPAQ we have 



OA = OP + PA = OP + OQ 

 Let OP = x and OQ = y we can write it 



a a = x + Vtt 



Multiplication. — We can keep the definition we have given for 

 directive magnitudes or we can even keep the definition of multi- 

 plication of fractions in arithmetic. 



From that definition we could give several representations of 

 the multiplication of complex quantities from which we will choose 

 the two following ones. 



First, Let us describe a circle of radius 1 with as centre 

 (Fig. 9), and let OX be our fixed direction, a a and bo the two 

 complex quantities to be multiplied. 



Draw OA = l a and OB such that 



XOB = a+'p. The angle XOB or rather 

 the arc XB is the argument of the product. 

 As arc XB = arc XA + arc AB we 

 see that a a . bp = (ab) a + p 

 represents a multiplication of tensors ab 

 combined with an addition of arcs of 

 circle a + /3 and as both these operations are commutative the 

 multiplication of complex quantities is commutative also. 



Fig. 9. 



Second, Otherwise let OA 



i 



Fig. 10. 



= a a OB = bp (Fig. 10) 

 Draw OP making with OB the 



angle a and take OP = OA x OB 



= ab. Then on OX take 01 = 1 



(01 = 1 is the unit). 



It is easily seen that the triangles 



OIA — OBP are directly similar 



and that OP = (a b) a + o 



