Interpretation of Quaternions, 115 



ordinate planes, and i n s j", k" will represent rotations of this 



line of intersection through angles = - in the planes of yz 9 zx 9 



xy respectively; and i l ,j' 9 k 1 similar rotations in the planes of 

 vox, wy 9 wz. In each case the origin of the rotations is deter- 

 minate. 



If the position of w be entirely arbitrary, the positions of 

 the intersections of the planes of wx, wy, wz with those ofyz 9 

 zx, xy, will be so also ; and the only difference arising in the 

 significations of i\f, k', i u ,j n , k", will be that the origin of 

 rotation is restricted only to the three coordinate planes suc- 

 cessively, the position in those planes being arbitrary. These 

 considerations will enable us to interpret the various terms in 

 the linear expression for Q; for 



ix — i{x, 0, 0, 0) = (0, x, 0, 0) . . . . (2.) 



Jy=J(2/9 0,0,0)= (0,0, y,0) .... (3.) 



kz = k{z, 0, 0, 0) = (0, 0, 0, z) ... . (4.) 



Now the first constituent of the quaternions on the right- 

 hand side of the above expression will, according to the prin- 

 ciples of interpretation above given, be considered as repre- 

 senting a line coinciding with the intersection of a plane pass- 

 ing through the axis of 



,r, with the plane of yz, in (2.); 



JA zx, ... (3.); 



z, ... ... xy, ... (4.); 



and consequently ix,jy, kz will represent that the lines whose 

 lengths are represented by x, y, z have revolved through 



angles each = - in planes perpendicular to their original di- 

 rections. 



Adopting the above interpretation of the various terms in 

 the expression for a quaternion, the question next arises, in 

 what sense are the lines represented by these terms, and by 

 quaternions generally, said to be added ? Now the funda- 

 mental formula for the addition of quaternions shows that in 

 whatever way the line Q is formed from the quantities w,x,y 9 z 

 (with similar expressions for any other quaternions Q 19 Q 2 . .), 

 then the sum %Q n is a new quaternion line formed in the same 

 manner from the quantities %r n , %x n , Xy n , %z n ; and writing 



2Q M =<St=(W, X, Y, Z), 



it appears that W will be the algebraical sum of the lines 

 ISO, w 19 . . , supposed, for convenience, to be similarly directed, 



12 



