286 



WILSON AND MOORE. 



For if s be held fixed and r take the values 1, 2, , n, there are ?i 



equations (21) which are linear and non-homogeneous in the w varia- 

 bles ,X'(*\ and the equations are consistent because the determinant 

 I jXr I of the coefficients does not vanish. If we replace jX'^'' and i\r 

 by their values from (20) we have 



Then 



2iS,a(*')..-X',2:„a,„.iX(") = e. 



Spga('-3)asp2i<„a(»'\iX',a™.iX(") = Spga^'-'^a^pe,, , 



Now by the reduction formula (13) the left hand side may twice be 

 simplified. On the right hand side e„ vanishes unless r = s and the 



double sum reduces to 



■PQ' 



Hence 



Si.iX'^.iX^'^ 



-PQ' 



(210 



We see therefore that there is a reciprocal relation (21') to (21) be- 

 tween the iX„ i\'r, iX'"-), iX'^'K 



The n systems Ar may be called a co variant n-tuple; the systems 

 tX^'') the contra variant ?i-tuple; these are mutually dual in pairs. 

 The set of n systems ^X'^''^ will be called reciprocal to the set iX^''^ 

 and the set i\'r reciprocal to the set i\r. We may give a geomet- 

 ric analogy in support of this nomenclature. If we have a conic 

 and three points P, Q, R, we may obtain the duals, the lines p, q, r. 

 The points, however, determine three lines QR, RP, PQ and of these 

 the duals are the points qr, rp, pq. The sets P, Q, R and qr, rp, pq 

 are reciprocal; and similarly p, q, r and QR, RP, PQ. Another 

 analogy would arise in spherical geometry where ABC and A'B'C 

 are polar triangles; the sets A, B, C and A', B', C, being reciprocal. 

 The use of reciprocal systems in vector analysis is prominent in the 

 system of Gibbs, particularly for the solution of equations. If the n 

 sets tXr form an orthogonal ?i-tuple, the reciprocal sets will be pro- 

 portional to them — a unit orthogonal 7i-tuple is self-reciprocal 

 (see infra, §13). 



• We may obtain in addition to the defining relations, the following 

 between reciprocal ?i-tuples. 



l^i.iK's.iK = ast, Si.iX'W.iXC^ = a(^'). 



(22) 



These are proved in the usual fashion. If we compare the relations 

 (21) which define the elements ik'^^^ in terms of the elements iKr with 



