VECTOR ANALYSIS. 65 



The criterion of a versor may therefore be written 



For the last equation we may substitute 



It is evident that the resultant of successive finite rotations is 

 obtained by multiplication of the versors. 



143. If we take the axis of the rotation for the direction of i, 

 i' will have the same direction, and the versor reduces to the form 



ii+j'j + k'k, 

 in which i, j, k and i, j', k' are normal systems of unit vectors. 



We may set 



j' = cos q j + sin q k, 



k' = cos q k sin q j, 

 and the versor reduces to 



ii + cosff {jj+kk} + sing {kj jk}, 

 or 



ii + cos q {I ii} + sin q Ixi, 



where q is the angle of rotation, measured from j toward k, if the 

 versor is used as a prefactor. 



144. When any versor <J> is used as a prefactor, the vector $ x 

 will be parallel to the axis of rotation, and equal in magnitude to 

 twice the sine of the angle of rotation measured counter-clockwise as 

 seen from the direction in which the vector points. (This will appear 

 if we suppose 3? to be represented in the form given in the last 

 paragraph.) The scalar <l s will be equal to unity increased by twice 

 the cosine of the same angle. Together, < x and <fr s determine the 

 versor without ambiguity. If we set 



^ = TT<P 

 the magnitude of will be 



... , n - or tan iff, 

 2 + 2 cos q 



where q is measured counter-clockwise as seen from the direction in 

 which points. This vector 0, which we may call the vector semi- 

 tangent of version, determines the versor without ambiguity. 



145. The versor $ may be expressed in terms of in various ways. 

 Since < (as prefactor) changes a 0Xa into a + 0Xa (a being any 

 vector), we have 



Again 



00+{I+I x 0} 2 _ (1-0.0)1 + 200+21x0 

 1 + 0.0 1 + 0.0 



G. II. E 



