182 7 G. PLARR ON THE ESTABLISHMENT OF THE 
in a second case on 
Tae : 
1, + jb + ke) for Up, 
and on 
Deane E 
ae (72 + jy + kz) for Ua; 
in a third case on any other set of expressions for Up and Ua respectively | 
must be transformable or reducible to one and the same typical result. 2d, The 
tensor of the expressions which have to represent the products Up Ua, Ua Up, 
must be equal to unity. 
These are two conditions the fulfilment of which will necessitate some 
developments. | 
As to the choice of the rule of multiplication, of course we make choice of 
the rule of the distributive law of algebraic multiplication, and of the rule of 
signs included therein, and we adopt, moreover, the ru/e concerning the order 
of the vector- (or versor-) factors, putting to the left place, in the partial 
products, those factors which belong originally to the multiplier. 
As to the means of fulfilment of the conditions, we have at our disposal the, 
as yet, undetermined meaning of the products, neve in number, 22, 77, 7k, 77, &., ' 
and of the three products 
Up Up, Up Uc, Uc Up. 
Their values and meaning, and mutual relations, will, by these means, become 
determined. 
Let us now treat of the first condition. 
We apply the distributive law, and the other rules, to the expressions— 
p= Tp Up 
w~ aN 
w = Taw (Up cos paw + Ua sin pa), 
and we provisionally put 
Zan YN 
g, = Tp Ta[(Up)’ cos pw + (Up Uo) sin pa] 
i= Tp ln(W ep) costeat Uc Up) amie: 
Again, we apply the same rules to— 
p=tat yb + ke 
aw=mu+jyt+ kz, 
and put 
$¢, = ala + yy + thz| 
+ b[ jiat py + jkz| 
+ c[hiv + hy + Pz] 
$,=a@[Va + yb + tke] 
} + y[jiat 7b + jke| 
+ z[kiat+ kjb+ ke]. 

