Sir Witt1am Rowan Hamitton’s Researches respecting Quaternions. 203 
which allows us to establish also this other symbolical equation : 
r 
Min 1,6, 1m ye (18) 
For example, if we take, in this last expression, the values n = 4, r=1, s = 2, 
we are conducted to the following characteristic of a certain transposition of the 
moments of an octad, which transposition, if it be once repeated, will restore 
those eight moments to their original arrangement, and which is therefore to be 
regarded as being a symbolical square root of unity ; namely, 
Tie (19) 
if 
© = M4,5,6,7.0, 1, 2,3. (20) 
It may also be here observed, as another example of the notation of the pre- 
sent article, that if, in addition to this last characteristic w, we introduce three 
other signs of the same sort, which we shall call (for a reason that will afterwards 
appear ) three coordinate characteristics of octadic transposition, and shall define 
as follows : 
@, = Ms, 0,7,2,1, 43,69 | 
@, = M6, 3,0,5,2, 7, 4,19 (21) 
3 = Mz, 6,1, 0,3, 25,43 | 
then these four symbols, w, ,, ,, 3, will be found to be connected by the rela- 
tions, 
2 
0, = 0, =, = 0, ©, 0, = 0; (22) 
WW, = WW; WH, = 0,0; WH, = WW; (23) 
from which, when combined with the equation 
op = I, (24) 
these other symbolic equations may be deduced : 
W, W, = W335 WW; W535 03,0, = 5 } (25) 
0, W, = WW,5 W, W, = WW,5 ww, = WH; 
W, W, W, = Wy WW, = W, W, W, = W; ] (26) 
W, Wy W, = W, W, W, = w, 0,0, = 1; J 
(ww,)” = (ww,)* = (ww,)? = w; 
(wo,)* = (ww, )' = (ww,)' =1; (27) 
Oo =o, =o, =I. 
