194 G. PLARR ON THE ESTABLISHMENT OF THE 
Uz, as to its positive part, is the same as that of V(pa), so that the direction 
of 
Upy Uc:, U(Ven) = Us 
form a triple rectanglar system arranged in respect to one another in the same 
manner as the directions of 7, 7, & respectively. 
It follows that by the application of the rule of multiplication to the forming 
of the product 
Wie; 
we get the result Uo ; because, jirst, the scalar of the product is zero, because 
the angle a: = 90°; second, the vector of the product is perpendicular to the 
directions of both factors, and its positive part is directed in respect to Uz, Up, 
in the same manner as Ur is directed in respect with Up, Uc; third, and finally, 
the product is equal to its vector, the scalar being zero. 
By actual multiplication, or by the generalisation of the result that 
(Up = fh =—1, 
in all possible directions we get also the results 
(Uc)? =—1 077" =—1. 
It follows from all this that the three vectors 
Vey Woy Un, 
relating to any triply rectangular system of directions, arranged in the same 
mutual arrangement as those of 
OS 96 k, 
satisfy the same relations as 2, 7, #, or, in other words, the relations between 
i,j, & are only a particular case of the relations— 
p” = oo” = 7? ——— 1 
APs p’ 
7p’ =— pr a 
p'o'=— o'p' = 7, f 
where p’, o’, 7’ represent respectively Up, Uo, Ur. 
We remark the formule 
assesh Via — V(po) x p 
Dr aye Pee T(Vpw) x To’ 

