20] 
123 
20. 
ON CERTAIN RESULTS RELATING TO QUATERNIONS. 
[From the Philosophical Magazine, voi. xxvi. (1845), pp. 141—145.] 
In his last paper on Quaternions [Phil. Mag. voi. xxv. (1844), p. 491] Sir William 
R. Hamilton has alluded to a paper of mine on the Analytical Geometry of (n) 
Dimensions, in the Cambridge Mathematical Journal [11], as one that might refer to the 
same subject. It may perhaps be as well to notice that the investigations there contained 
have no reference whatever to Sir William Hamilton’s very beautiful theory ; a more 
correct title for them would have been, a Generalization of the Analysis which occurs 
in ordinary Analytical Geometry. 
I take this opportunity of communicating one or two results relating to quater 
nions ; the first of them does appear to me rather a curious one. 
Observing that 
(A + Bi + Cj + Dk)- 1 = (A- Bi - Cj - Dk) (A 2 + B l + C 2 + D 2 ) (1) 
it is easy to form the equation 
{ A + Bi + Cj + Dk)- 1 (a + /3i + yj + 8k) (A + Bi + Cj + Dk) \ 
1 j 
~~ A 2 + B - + C 2 + D - 
r a. (A 2 + B 2 + C 2 + D-) ,(2) 
+ *'[ ¡3 (A 2 +R 2 -G 2 -D 2 ) + 2 7 (RG+AD) +2S(BD-AC) ] 
+j[2/3 (BC-AD) + y(A 2 -B° + C 2 -D 2 ) + 28(CD + AB) ] 
, +k[2/3 (BD + AC) +2y(CD-AB) + S (A 2 - B 2 - C 2 + D 2 )] J 
which I have given with these letters for the sake of reference ; it will be convenient 
to change the notation and write 
16—2