^} (41) 



498 On some Extensions of Quaternions. 



every vector is a scalar," which may be thus expanded : 

 tsr2= {i,x^ + . . tA)2= (l)a?,2 + (2)0^2^ + . . + («)a;„n ^ 

 + 2(12)^1*2 + 2(13)a;ia?3+ . . +2(/y)a:^a^^+ . .;/ 

 that is, more briefly, 



{•2,cx)^=z'2.{eW + 2^{fff)XfX„ .... (39) 

 the summations extending to values of the indices > 0, and g 

 being >/. In the second place, and inore generally, " inverted 

 products of any two vectors are equal to conjugate polynomes ;" 

 or in symbols, 



t!7''57 = K.-57< (40) 



whatever two vectors may be denoted by ct and ot'. In fact, 

 these two products have (according to the definition [3.] of con- 

 jugates) one common scalar part, but opposite vector parts, 

 S . ■or'CT=S . 'sris'='Z{e)a;eX'e + 'Z{fg){xja^g +XgCc'f) ;' 

 —V . '57''5r = V . ■ut'st' = l.{fgh){x^ g —Xgx'^)ih : 

 whence also we may write, as in quaternions, 



S .■0T'07' = i(OTOT' + '!!r'OT), V . •nr'cr'=i('B3-0T'— ■ct'ct). . (42) 



And, thirdly, the result (40) may be still further generalized as 

 follows : — " The conjugate of the product of any two polynomes is 

 equal to the product of their conjugates, taken in an inverted 

 order;" or in symbols, 



K.PF = KF.KP (43) 



In fact, we have now, by (16), (24), (27) and (40), 



KP" = K.PF=K. (j9 + XT)(y + 1!r') 

 = K(^' -{-pt^ +p'is + •croj') 

 =^pp^ — P"^ — p''^ + '^'^'^ 

 = (y-«')(i^-'=^) = KP'.KP, . . (44) 



as asserted in (43). It follows also, fourthly, that " the product 

 of any two conjugate polynomes is a scalar, independent of their 

 order, and equal to the difference of the squares of the scalar 

 and vector parts of either of them ;" for, 



if P' = KP, then PF = ( j9 + «r) ( j9 - w) =p^-zr'^ ; . (45) 



where w^ is, by (38) or (39), a scalar. And if we agree to call 

 the square root (taken with a suitable sign) of this scalar pro- 

 duct of two conjugate polynomes, P and KP, the common tensor 

 of each, and to denote it by the symbol TP ; if also we give the 

 name of versor to the quotient of a polynome divided by its own 

 tensor, and denote this quotient by the symbol UP : we shall 

 then be able to establish several general formula, as extensions 



