188 ° G. PLARR ON THE ESTABLISHMENT OF THE 
(Up at? =f=Kh=h, 
Up Uc = — Vo Up = gUz, 
jk=— kh =, 
a= — k=, 
as the consequence of f = 0. 
We now proceed to the treatment of the second condition, namely, that the 
tensor of the expressions which are to replace U(pa) and U(awp) must be equal 
to unity. 
This condition applied to ¢,, ¢,, gives for one of them— 
T¢, 
Tp To 

a Ties. 
= T[hcos pa + gU7sin po] = 1. 
As we have defined the tensor of a quaternion, the condition transforms itself 
into 
JX oN 
(h cos pa)” + (g sin pw)” = 1 
for both ¢, and 9,. 
The absolute values of g and f are equal to unity, because the tensor of each 
must be equal to the products of the tensors of two versors, and these latter » 
being equal to unity; therefore the squares 
i= +1, h°= +1. 
The above condition will now be satisfied by the admission of 
on = In =2: 
and finally we admit 
| pa =, 
wp = Q,- 
The definition of the tensor of a quaternion has now been particularised to 
being, (16) :— 
(Tq)’ = (Sq) + (TV9)’. 
§ 4. Determination of Y and b. Discussion of Results as to the Rule of Multiplication. 
The values of g and } are both to be chosen between either + 1 or — 1. 
As to g, we have to apply the rule given by Sir W. Rowan HamiLtTon 
regarding the sign to be attributed to the product of two versors (unit-vectors) 
perpendicular to each other. 

