190 G. PLARR ON THE ESTABLISHMENT OF THE 
there are more factors concerned, or in the case of the multiplication of quater- 
nions, would lead to a useless complication. 
So from the point of view of practical application we must determine the 
convenient value for J. 
We shall determine f so as to gain the applicability of the distributive law 
to the multiplication of two conjugate quaternions. 
Let w +, and w—a, be the two quaternions, w being a scalar, wa 
vector. 
Their product, indicated by (w + w)(w — @), is to be expressed according 
to our desideratum by 
R=w + ow— wo—o’, 
namely, by 
R= w’?—a’. 
The sole condition to be satisfied is, that the tensor of the expression R be | 
equal to the product of the tensors of the factors ; the versor of the expression 
may then become what tt may. 
Now we have 
T(w + @) = T(w — o) = 2/w*? + Tio)’ . 
Therefore we have the condition 
w* + (To)? = T(w? — o’) . 
But 
o = To Uo 
wo = (To)’(Ue)’ = b(To)’ . 
Therefore we have 
Tw? — wo”) = T[w’ — h (Te)’]. 
As the quantity of which the tensor is to be taken is a scalar (h being a scalar), 
the tensor is the quantity itself taken with the appropriate sign. Thus the con- 
dition becomes 
a + (Vo)? = [w — h(Te)?] f 
It is easily seen that the upper sign alone will make w’ disappear, and then we 
have the condition 
(1 + 5)(To/’ =0. 
Giving, for whatever values of w and To, 
b= — 

