

ELEMENTARY PRINCIPLES OF QUATERNIONS. 201 
Applying to the first member the principle of the associative property, we have 
for it 
Cx. (BEB) C xa)? 
so that 
Cx Bi = AL x KB. 
We may now divide by the scalar factor (TB)’, and replacing C by what it 
represents, A ~ B, we have the result of division— 
; 1 
A+B = app * A x KB. 
In the particular case of A being a scalar only, this formula gives : 
| 1 ines 
B or (B) = (TB 5 
We may now write the general result of division under the form 
A 1 
po Xe OS A x Be. 
Thus division of a dividend by a divisor is effected by making the product of 
the inverse of the divisor (as multiplicand) by the dividend (as multiplier). 
In other words, a fractional expression like A +B, or es may always be 
replaced by the product of the inverse of the denominator, multiplied by the 
numerator, namely, by 

This absolves us, to say once for all, from the consideration of quotients under 
the form of a fraction. 
We may add the general remark, that multiplication of an expression by 
another is expressed by writing the multiplier to the left of the expression 
| to be operated upon ; and that division of an expression by another is expressed 
| by writing the inverse of the divisor, as a multiplicand, to the right of the 
expression to be operated upon. 
When the scalars of A and B are zero, namely, when both quaternions 
| reduce themselves to their vectors, o and p suppose, then we have KB =— p, 
| and the quotient 
dk ! 
a == = Tp (cos po + Ursin po). 
| This is the typical expression which the inventor of quaternions took as the 
| starting-point for their theory. 
