60 VECTOR ANALYSIS. 



If a is a unit vector, 



{I X a} 2 =-{I-aa}, 

 {Ixa} 3 = IXa, 

 {Ixa}* = l aa, 



etc. 



If i, j, k are a normal system of unit vectors 



i = i xl = kj jk. 



= . 

 If a and /3 are any vectors, 



That is, the vector axft as a pre- or post-factor in skew multipli- 

 cation is equivalent to the dyadic {fta aft} taken as pre- or post- 

 factor in direct multiplication. 



[axft]Xp = {fta aft}.p, 

 px[axft] = p.{fta aft}. 



This is essentially the theorem of No. 27, expressed in a form more 

 symmetrical, and more easily remembered. 

 132. The equation 



aftxy + /5yXa + y axft = a.ftxy I 

 gives, on multiplication by any vector p, the identical equation 



p.aftxy + p.ftyXa + p.y axft = a.ftxy p. 



(See No. 37.) The former equation is therefore identically true. 

 (See No. 108.) It is a little more general than the equation 



which we have already considered (No. 124), since, in the form here 

 given, it is not necessary that a, /3, and y should be non-complan 

 We may also write 



Multiplying this equation by p as prefactor (or the first equation b; 

 p as postfactor), we obtain 



/9./3xya + p.yXa/3 -f /3.a 

 (Compare No. 37.) For three complanar vectors we have 



a/5xy + /3yXa + yaX/3 = 0. 



Multiplying this by i/, a unit normal to the plane of a, ft, and y 

 we have 



+ yaX/3.i/ = 



