Prof. Donkin on the Geometrical Theory of Rotation. 189 



of arcs on the sphere correspond to and define the effect of rota- 

 tions of a solid round the same axes, but through double the 

 angles, it follows from these theorems that the quaternion 



cos^ + sin 2 (z7 +>« + /:«) . . • • (1-) 



represents the positive rotation of a solid through an angle 6, 

 round an a>ds whose direction cosines are /, m, n referred to axes 



fixed in space ; whilst 



a a 



co^-— mi- {il+jm-Vkn) .... (3.) 



may be intei-preted similarly with reference to axes fixed in the 



Mr Boole, in noticing the interpretation of the fomier of these 

 expressions (Phil. Mag. vol. xxxiii. p. 279), observes that a qua- 

 ternion w + ix+jy + kz, whose constituents do not satisfy the 

 condition w' + x''-\f + z^=\, is not directly mterpretable m 

 geometry. This, however, is certainly not true with reterence 

 to interpretation by means of the rotation of lines ; for in that 

 case the quaternion aq (where q does satisfy the above condition, 

 and a is a numerical coefficient) represents (as I have shown) the 

 rotation of a radius vector, combined with an alteration of its 

 lenqth in the ratio of a to 1 . And I think it not impossible that 

 some coiTCsponding interpretation of the coefficient may be chs- 

 covered in the case of the soUd. , . i 



At present, however, I proceed to illustrate the subject by one 

 or two additional examples. ^ , . ^ 



In all that follows, ^rj^ will refer to axes fixed m space, and 

 xyz to axes fixed in the rotating body. . , , „ 



Let p q r have their usual significations m the theory ot ro- 

 tation, and «;2=/- + r/ + r2. Also let the position of the body 

 at any instant be defined by the values of X, ^, v ; where 



d e e 



X= I tan H, /^ = '« tan ^, v = n tan ^, 



and e is the angle through which the body would have to turn 

 round an axis whose direction cosines are /, m, n,in order to pass 

 from the position in which the two sets of axes coincide, into 

 that in which it actually is at the instant considered. To adopt 

 the names proi)osed by Mr. Cayley and Mr. Sylvester X/tv arc 

 the coordinates of the resultant rotation, and /, m, n are the direc- 

 tion cosines of the axis of displacement. Lastly, let 



8ec«^=l + X' + /^' + v' = «- 



