1898.] MILLER — ON THE QUATERNION GROUP. 317 



satisfied by the substitutions which correspond to the unities that 

 are employed.^ 



These relations between the quaternion unities could also have 

 been obtained directly by means of the corresponding substitutions. 



As any relation between quaternion unities remains true if we re- 

 place all these unities by those which correspond to them in any 

 simple isomorphism of their group to itself, it follows directly that 

 a knowledge of the group of isomorphisms of this group to itself 

 is of great utility in transforming quaternion relations j e. g., from 

 the simple isomorphism 



I I I 



— I I — I 



I ; J 



— i i —j 



it follows that / may be replaced by j, j by k, and k by i at the 

 same time. In other words, we may always perform the substitu- 

 tion ijk. ( — i) ( — /) ( — /^) on the three imaginary unities of quater- 

 nions. By means of this substitution we can obtain each of the 

 three relations given above from any one of the set. The twenty- 

 # four possible substitutions in these imaginary unities can be directly 

 obtained from the given group of isomorphisms of Q. They are 

 the following : 



ijk. (-/) {-j) (-k) j (-k) (-j) k ij. {-i) (-j). k (- k) 



-0- J (-J) i (-» (-^)- Jk (-0 jk {-J) {-k) i (-/). /k. {-j) (-k) 



-/). k (-k) ij i—k). k (—i) (— » ik (— /) {—k) i (-0. j i—k). {—j) k 



-j). k {—k) i {—j) k.j (—k) (-i) i (—k) (~i) k i (—j).j (_/). k (—k) 



ikj. (-0 i—k) (—j) i (—j) {—i)j ik.j i-j), {—i) {—k) 



ik (—/)• j (—0 (—k) ij (-0 i—j) i {-k).j {—j). k (-i) 



i{-k){-j)-j{-i)k 



i{-k)j,k{-j){-i) 



when an equation between the quaternion unities admits a of these 

 1 Cf Tail's Quaternion, 1890, p. 46. 



