196 G. PLARR ON THE ESTABLISHMENT OF THE 
which is the expression of the distributive law in its most general form as to 
vectors. 
The formula is easily generalised to the case of three terms, p = p’+ p”+ p’”, 
a=o+a +a’. 
Let us apply this to the squaring of a vector, \ + p + v. 
(A+ p+ v) + tv) = 4p + dy 
+ pd + pw? + py 
+vh+uptr’. 
Now we have 
Aw + pA = 2Sdrz , 
because 
Vpr\ =— Voy. 
Therefore 
(A+ p tv)? =N+ pt rv 
+ 2(Sdu + Svd + Spr). 
Multiplication of Two Quaternions, not Conjugates necessarily, one by the 
other. 
Let p=a+a,qg =) + B be the quaternions, where 
a= Sp >, b= Bg 
Oi ND aN Ge 
Our demonstration will consist in the verification of the distributive rule applied 
to (a + a) (6 + 8), and the test will be the condition, that the tensor of the 
expression taken as product, will have to be equal to the product of the tensors 
of the factors. 
Let us put 
ab + SaB =e 
ab + aB+ VaB=y, 
and let us verify if we have 
T(c + y) = T(a + a) x T+ B)? 
By what we have established in the instance of two conjugate quaternions, 
we have 
T(a + a) =a? — a’, &e. 
Therefore we have to verify if 
ef — y* = (a — a?) (O* = B’); a 

