of Four- Dimensional Vector Analysis. 395 



Again we should have : 



or i 2 h = hh. 



The product iii 2 hh will be denoted by I. 



When I enters into any expression it obeys the ordinary 

 rule of sign, i. e. an odd number of displacements of I in a 

 vector product changes the sign, while an even number 

 leaves the sign unchanged. 



Iaj3y = — nifty — oiftly = — ufiyl . 



Also It 3 t 3 = —iiii 



and L' 2 i 3 2 4 = tj etc. 



Thus the factor —I will change a product V 2 into its 

 reciprocal and V 3 is changed into a vector of the type Vj. 

 We have evidently 



IY 3 a/3 7 = V 1 I«/3 7 (4-3) 



§ 5. The definition of the component of a vector p along 

 any direction denoted by the unit vector n is similar to that 

 in ordinary vector analysis. We define the component to be 

 — V pn. 



In the case of a plane Ave define the component in the 

 plane of a vector of the type V 2 in a similar way. 



The component is — V TVj where v denotes a unit plane. 



Thus the component of P 2 in the plane yz is 



— VoP 2 ^3 = P^. 



We cannot define a component along a vector normal to 

 the plane since the normal is indefinite ; we may, however, 

 define a component in the reciprocal plane. 



Let V 2 a/3 denote a unit plane, and Y 2 'a/3 its reciprocal. 

 The component normal to V 2 may be defined as 



-V P 2 V 2 '«/3, 



or what is the same thing : 



V P 2 (IV 2 */3). 



This is equal to V IP 2 V>/3= -V P 2 'V 2 «£. 



Thus the component of a vector in the reciprocal (or 

 normal) plane is equal to the component of the reciprocal 

 n the plane. 



