26 VECTOR ANALYSIS. 



We may infer from the similarity of these equations that the relations 

 of a, /3, y, and a', /3', y are reciprocal, a proposition which is easily 

 proved directly. For the equations 



Q_ yxcc ax 



* & . ? > P ~& 'TT ? > y ~ / v . >/ 



a.pxy p.yXa y.aXp 



are satisfied identically by the substitution of the values of a', /3', 

 and y' given in equations (1). (See Nos. 31 and 34.) 



Def. It will be convenient to use the term reciprocal to designate 

 these relations, i.e., we shall say that three vectors are reciprocals of 

 three others, when they satisfy relations similar to those expressed in 

 equations (1) or (4). 



With this understanding we may say : 



The coefficients by which any vector is expressed in terms of 

 three other vectors are the direct products of that vector with the 

 reciprocals of the three. 



Among other relations which are satisfied by reciprocal systems 

 of vectors are the following: 



a./3' = 0, a.y' = 0, /3.a' = 0, /3.y' = 0, y.a' = 0, y./3' = 0. (5) 



These nine equations may be regarded as defining the relations 

 between a, /3, y and a, /3', y' as reciprocals. 



(a./3xy)(a'./3'xy)=l. (6) 



(See No. 34.) , , Q Q , , , A 



aXa +/3x/3+yXy =0. (7) 



(See No. 29.) 



A system of three mutually perpendicular unit vectors is reciprocal 

 to itself, and only such a system. 



The identical equation 



P = (p-i)i+(p-j)j+(p.k)k (8) 



may be regarded as a particular case of equation (2). 

 The system reciprocal to aX/3, /3xy, yXa is 



a'x/3', /3'xy', y'xa', 

 or a /3 y 



a./3xy' a./3xy' a./3xy* 



38a. If we multiply the identical equation (8) of No. 37 by <rXr, 

 we obtain the equation 



'.cra.T y.ra.o-) + y./o(a.<7/3.T a.r/3.0-), 



which is therefore identical. But this equation cannot subsist identi- 

 cally, unless 



