192 G. PLARR ON THE ESTABLISHMENT OF THE 
As an application of therule of multiplication, let us determine Uc, know- 
ing that the vector Vpa of the product of two vectors p, a, gives the perpen- 
dicular to both, its positive part being determined according to the rule above 
explained. 
Let us take as factors Ur, Up, in respect to which Uo is directed as Ur 
was in respect to the factors Up, Uc, then we have Uc=V(U7Up). 
Now Uris determined by 
A 
V(pa) = Tp To sin po Ur 
/N 
T(Vpo) = Tp Tasin pa , 
and as Vpza is also 
V(pa) = 1a, + 7b, + ke, , 
where we put 
a, = bz — cy 
b, = ca — az 
C= ty = OF, 
we have 
i : : 
Ur= TVpe [ia, + 7b, + ke, | 
Up= tp lia + jb + ke]. 
And then we must have 
x (ity +) 0g + Keg) a + 7b + ke) 

Making the product, and calculating it by its scalar and its vector, we have for 
the scalar, in virtue of 
P=pPp=hP=—1, 
the expression whose numerator is 
— (a,a + bb + ¢,¢) , 
and this is = zero by the very expressions for a, 6,,¢,, aS it ought to be, Uc 
being a vector or unit vector. 
The vector part of the effected product, by the application of 
jk= —hj =i 
w= th=7 
y= —pZk 

