114 Mr. W. Spottiswoode on the Geometrical 



In accordance with the fundamental idea of a quaternion, the 

 position of the line represented by any constituent will be sup- 

 posed to depend upon the position which that constituent 

 holds in the first expression for Q; so that the directions of 

 the four lines being once chosen, the first, second, third, and 

 fourth constituents will always represent lines drawn in the 

 four directions respectively, whatever changes may have taken 

 place in the order of the constituents as originally given. Now, 

 returning to the equations (13.) and (14.) of the former sec- 

 tion, it appears that i" indicates a change by which the nega- 

 tive axis of z is brought into the old position of the axes of y; 

 and the axis of y into the old position of the axis of z; the 

 positions of \v and x remaining unchanged ; or i" may be con- 

 sidered as indicating a change by which w and x are brought 

 into positions such, that they are situated, with respect tot he 

 negative axis of z and the positive axis ofj/, in the same man- 

 ner as they were at first with respect to the axes of 3/ and z 

 respectively. When the axes are rectangular, as at present, 

 this change may be represented, either by supposing the plane 

 of yz to revolve in its own plane through half a right angle in 

 the direction from y to ss 9 or by supposing the three axes to 

 remain fixed, and the radius vector w to revolve on the surface 

 of a right cone with a circular base, whose axis is that of x 

 and vertex the origin, through one quarter of a revolution ; 

 the direction of rotation being from the axis of z towards that 

 of y. It is easily seen that j" and k" may be represented by 

 similar revolutions about the axes of y and z respectively. 



Again, i' indicates a change by which the negative axis of 

 x is brought into the old position of w, and w into the old 

 position of the positive axis of x, the positions of y and z re- 

 maining unchanged; and if a be the angle between the axis 

 of x and the line w (the vertical angle of the first cone), this 

 change may be represented by bringing the negative axis of 

 the cone into the old position of w, and then opening the ver- 

 tical angle of the cone through an angle = tt— 2a. The 

 changes j' and k' may similarly be represented by supposing 

 the negative axes of the other two cones respectively to take 

 the position of w, and the vertical angle of the cones to be 

 opened through angles =7r — 2/3 and 7r—2y respectively. The 

 above theory becomes much simpler when the position of w is 

 not absolutely determined, but merely restricted to a given 

 plane; in this case its position may be supposed to coincide 

 with the intersection of that plane with one of the coordinate 

 planes ; e. g. in the case of i, with the intersection with the 

 plane of yz ; in that of j\ with that of zx ; and in that of k, with 

 that of xij\ the three cones then become simply the three co- 



