ELEMENTARY PRINCIPLES OF QUATERNIONS. 185 
But the two first expressions are evidently symmetrical about both sides 
of— 
ay) 
a " D/ 
C J 
v 1 
v t 
C D 
Jf . 
CC’, and the two last about DD’. Therefore, if 7 + 72 = 2f” contains a vector, 
that vector must be directed in the direction either of +4, or of —%. But 
neither can take place, because the two first of the expressions are symmetrical 
in respect to the plane of the directions ¢, 7, through a common origin. We 
must conclude that f” cannot contain any vector part. 
The same mode of reasoning will apply to the expressions 2F, 2f’, &c., so 
that these four quantities must be scalars. 
The reasoning founded on the symmetry of the expressions which define 
F, f’, &c., does not apply to the four expressions in which g is concerned. We 
have, as for example— 
qh a ij i yi a ij ie Wavy + ji eu Uy + ji 7 


which shows that 29h relates to two consecutive quadrants, and has therefore a 
rotatory character. 
Further, we prove that 
Raf =f" =f” = tf suppose. 
Namely, by 7 — jt = gh, and by y + ji = 2f” we have 
y — f” av gh ; ; 
The second member is a quaternion, of which we know the scalar f”, and the 
vector gk; f’”, g, being admitted to be scalars. Therefore the tensor of 7 may 

