of the Concomitants of Two Quadrics 199 



Correlatively, S, %, x, S' are the four quadrics in u which make 

 the system of contra variants together with 



j = a^haUaU^ i-^B) {Ahu) {Ban). 



This latter represents the four vertices of the same tetrahedron. 

 In fact, the jacobian of u^, u^, (^Ahuf, (Bau)" is 



(a/3 Ab Ba) Ua,u^ {Abu) {Bau), 



where A = a a", say ; expanding the first bracket this becomes 



iijd^" (bBa) M - d/ b^ {ci"Ba) M + &„d/ {ci"Ba) M, 



where each term represents two, with a, a" permuted, and M is 

 short for UaU^{Abu){Bau). But the factor Ua is reducible to Ua' 

 (Gordan, il, § 6) ; which means in this case that the symbols u of 

 the factors u^,, (Abu) would be bracketed. Hence the product in- 

 volving cia is zero. Thus the jacobian is equal to 



baCip'(d"Ba)M, 

 = b^a^ {a"Ba')M+ba.h' {a"ab"a) M (if B = b'b") 

 = — ba a^ (a'a'B) M (as before) 



A correlative reduction applies to the case of /. 



The complexes. 



§ 7. A complex is a function of ]), or line coordinates, but not 

 explicitly of ii or x. There are eight quadratic and eight cubic 

 complexes in the system. The quadratics are 



(ApY or n, (Bjyf or TI', (abpf or tt,.,, (a/3p)' or IIi., 

 {AB){Ap){Bp) or C, {abp){a^p)a^b„., F;' and Fi. 



Differentiation. 



I 8. Let p be any symbolic product belonging to the whole 



dP 

 system; then — - {i=l, 2, 3, 4) would be composed of terms each 



OXi 



with one odd S}?mbol Qj or 6,: left over. Thus the four symbols 



^:— may be considered as the coordinates of a certain plane. For 



dxi ^ ^ 



example the coordinates of the polar plane of a point (x) with 

 regard to a^- are («a;«i, axU.,, a^a^, a^ai). Likewise ^ — would give 

 a set of point coordinates. 



