1 36 On the Equation Q = q{w, <r, 3/, z)=*iio-\- ix -{-jj/ + kz. 



upon the nature and properties of these functions, is entirely 

 without interest, I have ventured to suggest another mode in 

 which the question may be viewed. 



It is known that the symbol R„, when prefixed to any qua- 

 ternion, indicates that, out of the four constituents w, a-, 3/, ^, 

 regarded in a definite order, that one is selected which stands 

 in the wlh place from the left-hand. If, then, to the complex 

 symbol R„Q there be prefixed another symbol of selection R^, 

 it is clear that, since in the expression R^Q there is only one 

 constituent from which the new selection is to be made, the 

 combination R„jR„Q will vanish for all values of m, except 

 OT= 1. Hence may be formed the following symbolical system: 



R%=Ro, RiRo=0, R2Ro=0, R3 

 RqRj = Rj, R^j=:0, R2Ri = 0, RgRj^^ 



3R,=o I (,^j 



RoR2=R2, R,R2=0, R\=0, R3R2=0 



I^0^3=^3» RiR3=0, R2R3 = 0, R^g^O, 



and consequently, by the principles of the calculus, 



Q=(RoQ, R,Q, R2Q, R3Q) . (2.) 



= {(R%-R^-R22-R23)Q, -^ 



(R,Ro + RoRi + R3R2-R.R3)Q i (3) 



(R2Ro+RoR2+RiR3-R3Ri)Q ^ • • • • ^ •'' 



(R3Ro+RoR3 + R2R,-RiR2)Q} 



= (R%Q, R,RoQ, R2R0Q. R3R0Q) 1 



+ (R_iR,Q, RoRiQ, R_3R,Q, R^^iQ) I u) 



-f- (R_2R2Q» R3R2Q) RflRgQ, R-iRgQ) 



XR.sRgQ, R-^RsQ, RiR3Q» RoR3Q)» 

 =Ro.i,2,3RoQ+R-i,o,-3,2^oQ -1 



+ R-3.3.o.-iRoQ+R-3,-.,,,oRoQ/ • • ^""'^ 



—w + ix+j7/+kz (6.) 



The same result might have been obtained by means of the 

 relation 



(w, A-,3/, z) = (w, 0, 0, 0) + (0, a?, 0, 0) + (0, 0,3/, 0) + (0, 0, 0, z), (7.) 



the second side of which might be at once replaced by (4.); but 

 in some respects the former method is preferable. In either case 

 it appears that the expression (6.) is only one out of an infinite 

 number which might have been obtained, by substituting for 

 the expressions in (3.) any arbitrary combinations of Rq, Rj, 

 Rgj Rs* respectively equivalent to those symbols themselves ; 



