110 Mr. W. Spottiswoode on the Geometrical 



quantities, i. e. associated with nothing but itself; in other 

 words, it is simply equivalent to the ordinary algebraic quan- 

 tity w; so that by means of the law of addition of quaternions, 

 it will be allowable to write, 



(w,^3/,^)==(w,0 5 0,0) + (0,^,0,0) + (0 5 0,3/,0) + (0,0,0,s)\ 

 = W+(0,#,0,0) + (0,0,y,0) + (0,0,0 a s) J 



With respect to the last three terms of this expression, it will 

 be necessary to introduce some new symbols. Thus, for in- 

 stance, if T, T', T" indicate the operations of transposition 

 defined by the following equations, 



T (^0,0,0) = (0,^,0,0)~ > | 



1%, 0,0,0) = (0,0,^0) }>,... (9.) 



T"(z, 0,0,0) =(0,0,0,«) J 



the equation (8.) might be written 



Q=w + Tx + Ty + T"z. .... (10.) 



And, if the laws of the combination of the symbols T, T', T" 

 were known, the general laws of the combination of quater- 

 nions would be at once deducible. 



It will however be more advantageous to use some symbols 

 of transposition rather different from those above noticed ; let 

 then 



jQ=(-y, *,«>,-#) y, . . . . (ii.) 



from these definitions of the symbols of transposition, i,j\ k, 

 it is easy to deduce the following relations: 



iJQ=j.jQ = k.kQ==i.j.kQ = (-w,-x,-y,--z)=:--Q~ 

 j.kQ=-k.jQ=iQ 

 Jc.iQ=— i . kQ =jQ 

 i.JQ=-j.iQ=kQ 



y(12. 



in which the expression for — Q may be deduced from (5.) by 

 writing —1 for n. These relations may be symbolically 

 written, as follows: 



i 2 —j 2 = k 2 = ijk = — • 1 

 jk=—kj=i 

 ki= —ifc=j 



ijzsz-ji = k 



L . . . . (13.) 



