ELEMENTARY PRINCIPLES OF QUATERNIONS. 181 
are admitted to represent two different expressions, not equal to one an- 
other. 
This principle does not apply to the product of a vector by a scalar, or vice 
versa, and also it may be easily shown that if a scalar w is engaged as a factor, 
concurrently with vector factors, the place of the scalar may be shifted to any 
place in the product, namely, that there is equality between 
Was = ab = abw. 
We apply to pw, and ap respectively, the principle of decomposition into 
two factors, a tensor and a versor. The versor is simply the quantity pao (or 
ap) itself divided by its tensor (which is a scalar). That is, we put 
po = T(pa) x Ufo). 
The consequence is, that the absolute value of U(pa) is unit— 
PWiea): = 1. 
As we have also 
po = Tp Up Xx Ta Ua, 
it follows that 
Mi(pa) = Loox Ta; 
and 
U(pa) = Up Ua, Ulap) = Ua Up. 
The first equation is the expression of a general principle, which is applicable to 
any species of factors, namely, that the tensor of the product is equal to the 
product of the tensors of the factors. By an abbreviation we may call it the 
“law of the tensors” in a product. 
The two other equations give for the tensor of the first member the value 
unit, because we have defined the tensor of a versor, or unit-vector, to be equal 
to unit, and by the application of the principle about the tensor of a product. 
When Ua, or both Up and Ua, are replaced by expressions of a polynomial 
form, then we have a priori no indication of how the products are to be deve- 
loped into a sum of partial products, because the partial factors differ in direction. 
We shall be in need of a rule for effecting multiplication, whenever such 
expressions of a polynomial kind are introduced. 
The choice of the rule stands at our free will, but in all cases we have to 
satisfy two distinct conditions, namely—1st, The expressions which will have to 
represent the products Up Uwand Ua Up, and which will be based on different 
expressions for Up, and for Ua, [say in a first case on 
. A A 
Up and on (Up cos pw + Us sin pa) for Ua; 
