(25.) 



Interpretation of Quaternions. 113 



+.(o,*,^, f xr).(o,fi;o;6) 



+ (0,#,^s). (0,0, j/ l? 0) 



+ (0,^,3/,^). (0,0,0,^) 

 — (ww 15 w^ + a?^, Wiy+y{w 3 w^ + ^^ + iO^y^.^x^y^. 

 But if we adopt the form (18.), the constituents of the product 

 of two quaternions are completely determined ; in fact, it is 

 found without difficulty that if 



w 2 = wWj — ■r.Tj — ,3/j/i ~ «»! ^j 

 /r 2 =w?o^ 1 + t<o x x -\-yz l —y& I 

 y 2 =wy l \+w 1 y+zx 1 -:z 1 x f" 



^ 2 = W^! -f »!« + a^! — A'^ J 



And also, if 



Q 1 Q=Q 2 1 = < + i> 2 1 +jf> 8 l + ^ 2 l , . . • (26.) 



©a-^J «2=-4 J/2--J/2 1 ' 2 2="^' ' ( 27> ) 



Moreover 



(fay+j^+.**)*==.-** 8 -/- ; « a . • • • ( 28 -) 



Q*=w*-x 2 -y*-z 2 -r2w{ix+jy + kz) . (29.) 



(w — ice —jy — kz)(w + ix +jy + kz ) = w 2 + # 2 +# 2 + ^ 2 "V , , 



= (w? + i> +jy + kz)(w—ix —jy —kz). J 



So that the reciprocal of a quaternion is the quotient of the 

 quaternion itself, with the signs of its last three constituents 

 changed divided by the sum of the squares of the constituents. 

 The constituents of the ratio 



Q!=Q _1 Q 2 



may be found either by solving (25.) with respect to.u> 19 x l9 y v z v 

 or by means of the relation (30.), and so reducing the division 

 to multiplication. 



§ 2. Geometrical Interpretation. 



In the general expression 



Q = (w, x,y, z)=w + ix+jy + kz 9 . . . (1.) 



let tt>, x 9 y 9 ss represent straight lines drawn in several direc- 

 tions from the origin, and let x 9 y, z coincide with the three 

 positive axes of coordinates respectively, while the direction 

 of w is arbitrary ; x, y, z may then be considered as the co- 

 ordinates of some point, in general not the extremity of w. 

 Phil. Mag. S. 3. Vol. 37. No. 248. August 1850. I 



