112 Mr. W. Spottiswoode on the Geometrical 



with other like formulae, the distributive character of the sym- 

 bols i 9 j 9 k is also established. 



The following verifications, although not essential to the 

 theory, are perhaps worth noticing. If we had taken (18.) as 

 the definition of a quaternion with (13.) as the definitions of 

 i\j, k, we should have found 



i Q = iw — x + ky —jz 1 



jQ=jw — kx—y + is K; . . . . (20.) 



kQ == kw -\-jx — iy—z] 



so that the equations 



(qj = tall — * c *<2 = •"> 



which obviously involve also 



jQ=jQ l= jQ 2 = ... I .... (21.) 



kQ=kQ l = kQ 2 =...j 



give rise to the equations (3.) ; for i,j 9 h being symbolical ex- 

 pressions for ( — )*, render all real terms, to which they are 

 prefixed, imaginary in the ordinary sense of that word. The 

 same definition of a quaternion gives 



2Q„=2»„ + Sx B+> ;-% )( + ^ i , . . . (22.) 



which is in fact identical with (4.). 



But the principal advantage of the linear form of the ex- 

 pression for a quaternion is found in the processes of multipli- 

 cation and division. In the form (1.) it does not seem possible 

 to obtain a complete solution of the problem of multiplication ; 

 the following however would be the initial steps to such a 

 solution : 



Q . Qj = (w, x 9 y 9 z) . (w l9 x l9 y l9 zj 

 = {w{wvx v tt v z } ) 9 *[w v * v y v zj 9 y{^v^v.Vi^i), *(WiS*«Ste*i)} 

 = {w{p 19 x 19 y 19 ^) 9 9 9 0} 

 + {0 9 x(w 19 x ]9 y l9 z l ) 9 9 0} 



+ {0,0 9 y(w l9 x 19 y ]9 z 1 ),0} 



= (w ] w 9 vo x x 9 WrfJ, w^) 

 + {0, xw x + ,r(0, x l9 0, 0) + x(0, 0, y l9 0) + #(0, 0, 0, *,), 0, 0} 

 + {0, O.yw, +y(0, x l9 0, 0) +y(0 9 9 y l9 0) +y{Q, 0, 0, z,) 9 0} 

 + {0, 0, 0, z^ Y + z{0 9 x l9 0, 0) +3(0, 0,y l9 0) + *(0, 0, 0, jrj} 



