Interpretation of Quaternions. Ill 



the operations of transposition and change of sign being in- 

 dependent of the subject of operation. 



As it will assist the geometrical interpretation of the ope- 

 rations i 9 j, k hereafter, to separate each of them into two di- 

 stinct operations, the formulae to which such separation gives 

 rise may be properly noticed here. Let then 



i'Q=(-x 9 w 9 y 9 z) i"Q = (w 9 x, —z 9 y)~) 



k'Q=( — z 9 x, y, w) FQ=(«), —y, x, ss)J 

 there will then result 



i = i<i»=i»i "1 



j=fj"=fj' }, (15.) 



k = k'k" = k"k'J 

 to which may be added, 



iy = W", fk'^i'j", k»i ! =j'k",l 



fk"=:kH" = i"j" I, . . (16.) 



P(ii l i") = V(jj l f) = F{kk'k") = - 1 J 



where P represents the symbolical product without reference 

 to order. By means of the above properties of i,j, k, it will 

 be possible to transform the expression of a quaternion (1.) 

 into another of the same form as (10.) ; for 



(0, x, 0, 0) = i (x, 0, 0, 0)=ix 1 



(0, 0, y, 0) =j {y, 0, 0, 0) =jy I ; . . (17.) 



(0, 0, 0, as) = k{as 9 0, 0, 0) =h% J 



so that (8.) may be written thus, 



Q = >w + ix+jy + kz 9 (18.) 



in which i,j, k may be combined according to the laws de- 

 fined by (13.). It may be observed, that, since by means of 

 the condition (4.) the addition of quaternions is reduced to the 

 addition of ordinary algebraical quantities, the order and clus- 

 tering of the terms in (18.) is indifferent, so that the associa- 

 tive principle of addition among those terms is completely 

 established ; the same is obviously the case with respect to 

 the addition of quaternions in general. It may be further 

 remarked, that, since by means of (4.), 



itQ^XiQ,, JZQ n =ZjQ n , £SQ,=2KJ„1 



jkS,Q n =2,jkQ n , fo'SQ„=2HQ„, ySQ B =SyQ L (19.) 



=i2*Q„ =&-Q„, =*-2/Qj 



