232 Sir Witr1Am Rowan Hamrton’s Researches respecting Quaternions. 
MoO; Ny gies Ayes Mess } (164) 
20,0 =8§3 eo = ee oles i es 
the equations of detachment, included in the general formula (160), will then, 
by (161), be the sixteen following : 
(t = 0) (= 1) 
(ss =0) eate'c=aa+ab; ea +e'c’=ad +a'l’; \ (165) 
(§=1) eb+ed=ba+b; eb’ +e'd’=ba' +); Gr ==0, 4 =0) 
(§=0) fatfire=ca+ec'b; fa +f'c’=ca +c’; | (166) 
(§=1) fb+fd=da+d'b; fl+fd=ddi' +d’; J (i =0,a.—4) 
(ss =0) gat+g’c=ac+a'd; ga+g'c'=ac' +a'd; \ Shel LOT 
(.=1) gh+e’'d=lbe+bd; gh'+eg'd/=be +0'd'; GaN fea) 
(ss =0) hathc=cco+ecd; ha'+h'c' =ce'+ cd; (168) 
(§=1) Ab+Nd=de+d'd; hb'+Wd=de'+dd. } r= 1) 
Now the twelve equations (165) (166) (167) are all satisfied, independently 
of c, ec’, d, d’, if we suppose 
Ci ah, Os ee Og (169) 
and then the four remaining equations (168) take the forms, 
ha + (h’'—c)c=c'd; (h’—c-—d)c'=0;  } 
170 
(h'—c—d’)d=0; hat+(N-d')d=cd; J Cy 
which are satisfied by supposing 
Wi=e+d; ha=cd—-cd. (171) 
Accordingly, with the values (169), the sign of derivation Xx, reduces itself to the 
ordinary numeric multiplier a, so that we may write simply, 
ASS 5 (172) 
and while the other sign of linear derivation x, retains its greatest degree of gene- 
rality, consistent with the order of the sets, namely, couples, which are at present 
under consideration, so that the four numerical constants ¢ c’ dd’ remain entirely 
unrestricted, the symbolic equations of the form (153) become now, by (164), 
(169), and (171): 
