Extensions of Quaternions. 49 



which give also, as a consequence, this other relation : 



tri!+'^(r*+rj( r a+ r iJ=.-*iW • • ( 177 ) 



and then the other conditions of association will all be satisfied, 

 if we make, instead of (175), 



(B 2 ).. « 1 = ^ = Cl = 0, « 4 =(r, + r 2 + /- 3 ) 2 : . . (178) 



this system also involving five arbitrary constants, for example 

 s^qt^t^t^. The assertion respecting quines, which was made near 

 the end of [16.], has therefore been fully justified. 



[28.] Finally, as regards the system (A) of quines, it may be 

 observed, — 1st, that in this sijstem, by (162) and (175), we have 

 generally, 



(A 3 ).. SW = 0; (179) 



or that " the product of any two quadrinomial vectors tzr, ct', 

 reduces itself to a pure vector;" and 2nd, that, by (163) (165), 

 " this vector product, otct', is of trinomial form, involving no 

 part with i 4 for a factor." This product is therefore already seen 

 to be of the form 



W = tl X l + i a X 2 + £ 3 X s ; .... (180) 



but I say, 3rd, that " its three coefficients, or coordinates, X„ 

 X 2 , X 3 have constant ratios," or that "the product otot' may be 

 constructed by a right bne in space of which the direction though 

 not the length is fixed," and which may therefore be conceived 

 to " coincide in position with one fixed axis (£) of the system." 

 In fact, by (163) (165) (174), we have 



m 1 Xj = ?» 2 X 2 =?» 3 Xg=Xj .... (181) 

 and therefore 



(A 4 ).. W = X£ (182) 



if we make for abridgement 



fiBSyBjJVa-Vj) + Vi^i -~x { z ) 3 )+7n l m i 



(Ag) . .^ + in x r l U l x' 4 —x 4 x\)+?n^{x i .z' 4 —x 4 x' q ) + m 3 r a 



{x s z' 4 -x 4 x J 3 ), (183) 



and ^=?n~h l + m~h 2 + m~h 3 .... (184) 



In the 4th place, " if any quadrinomial vector ct be multiplied 

 by or into the axis £, the product vanishes;" or in symbols, 



(A,).. fr = 0, ^=0; (185) 



because by (172) the scalar coefficient X becomes — 0, if we 

 change either a-,, a 2 , a 3 , and x 4 , or a.',, x', 2 , a.'.,, and a'.,, to flip 1 , 



Pkil. May. S. 1, Vol. 9. No. 56. Jan. 1855. E 



