184 G. PLARR ON THE ESTABLISHMENT OF THE 
But we have also for Ur the expression— 
. “N : “w~ tes 
Ur =12 00s ar +7 cos yt + £ cos ar. 
We see now that the condition (I.) will be identically fulfilled if we establish, 
by definition, (13), the following four relations :— 
UpWe— Uap — 24 Va 
jk —kj = 22 
ki — th = 297 
j—ji = Ogh, 
where g is the same in the four equations. 
The most general expression for g would be a quaternion. But if g were 
to depend on the particular undetermined direction of the vector of the 
quarternion, then the solution of our problem would become indeterminate, and 
a practical solution might, in all probability, be despaired of. 
We avoid this generality by introducing a condition which is analogous to 
that which we have already agreed to in respect of the expression for a single 
vector—the agreement, namely, that the expression of a vector shall be inde- 
pendent of the absolute position of the vector, a condition which is of practical 
use in the establishment of the analytical addition of vectors. 
In the instance of multiplication we render its analytical operation practi- 
cally possible by introducing the principle, that when similar systems of direc- 
tions (like Up, Uc, Uz) are to be bound by a mutual general relation, that 
relation must exist, and be expressed independently of the absolute direction of 
their system. 
We therefore agree that g does not depend on direction, and therefore must 
not contain a vector part, nor any angle in the expression of its scalar. There 
remains but the agreement that, (14), 
gy = a numerical quantity. 
Let us consider equation (II.) In the case of the relations defining F, f’, 
&c., we cannot apply the principle laid down just now, because these quantities 
are not bound, like g, to enter into equations of definition (yk + fy = f’, &c.) 
by one and the same value, and they can vary from one system to the other, 
a prior. 
But we are able to establish, by reason of the symmetry of the expression of 
F, f’, &c., that these quantities cannot contain a vector part. 
As for example, let — 7, —j, be represented by 7’, 7’; by the rule of signs 
agreed to, we have then 
Yt =t7 +74, 00d e7 ILS Ute. 




