VECTOR ANALYSIS. 67 



Therefore, 



a.. a. 



a.ftft.y 

 (See No. 38.) That is, 



03 + 0i- 



a _ax/3./3xy , a.y 



AlSO, CN . t7 2 = --- nTo ---- * -- /T/0> - 



a.ftft.y a.pp.y 



Hence, 1 X0 2 = 2 -(l-0r0 2 )03+0i> 



_0 1 + 2 +0 2 X0 1 



* 8 ~ 1-M 



which is the formula for the composition of successive finite rotations 

 by means of their vector semitangents of version. 



147. The versors just described constitute a particular class under 

 the more general form 



aa + cos q {ft/3' + yy} + sin q [yft - By] , 



in which a, ft, y are any non-complanar vectors, and a, ft', y their 

 reciprocals. A dyadic of this form as a pref actor does not affect any 

 vector parallel to a. Its effect on a vector in the ft-y plane will be 

 best understood if we imagine an ellipse to be described of which 

 ft and y are conjugate semi-diameters. If the vector to be operated 

 on be a radius of this ellipse, we may evidently regard the ellipse 

 with ft, y, and the other vector, as the projections of a circle with two 

 perpendicular radii and one other radius. A little consideration will 

 show that if the third radius of the circle is advanced an angle q, its 

 projection in the ellipse will be advanced as required by the dyadic 

 prefactor. The effect, therefore, of such a prefactor on a vector in the 

 ft-y plane may be obtained as follows: Describe an ellipse of which 

 ft and y are conjugate semi-diameters. Then describe a similar and 

 similarly placed ellipse of which the vector to be operated on is a 

 radius. The effect of the operator is to advance the radius in this 

 ellipse, in the angular direction from ft toward y, over a segment 

 which is to the total area of the ellipse as q is to 2-Tr. When used as 

 a postfactor, the properties of the dyadic are similar, but the axis of 

 no motion and the planes of rotation are in general different. 



Def. Such dyadics we shall call cyclic. 



The Nth power (N being any whole number) of such a dyadic is 

 obtained by multiplying q by N. If q is of the form 2?rN/M (N and 

 M being any whole numbers) the Mth power of the dyadic will be an 

 idemfactor. A cyclic dyadic, therefore, may be regarded as a root of 



