150 
Proceedings of the Royal Society of Edinburgh. [Sess. 
U and are contragredient with the variables X. Under either definition, 
we have a permanent relation 
^1^6 + ^ 2^5 + P3P4 = ^ • 
But the introduction of these variables has another consequence : it 
necessitates the introduction of other variables. Thus we must retain the 
point- variables of the intersection of the line, having , . . ., Pg for its 
variables, with the plane, having , . . ., for its variables ; these can 
be taken in the form 
Xf'= U2P3-U3P2 + U4P6. 
+U3P^-|-U,P,, 
U1P2-U2P1 +U4P,, 
V=-U,Pg-U,P,-U3P,, 
being, of course, cogredient with the variables X. Next, we must retain 
the variables of the plane which passes through the point, having 
Xj , . . ., X4 for its variables, and through the line, having P^ , . . ., Pg for 
its variables ; these can be taken in the form 
U/= X,P,-X3P, + X4P,, 
U/= -X,P, +X3Pg-fX4P2, 
U3"= X,P,-X,Pg +X,P3, 
U/= -X1P1-X2P2-X3P3, 
being, of course, cogredient with the variables U. Next, because we have 
two points X and X", we have a line joining these points ; its coordinates 
can be taken to be 
X3X/', 
Pf ^X^X 
p/=X3xy-x,X3", 
P " _ V “V " V V" " 
3 — ^2 ^ 2^1 5 
P/ = X,X/.-X/'X,, 
P/ = X„X/-X/X,, 
P ff V V " v "v 
4 — -^3-^4 ^3 ^4 , 
with the relation 
Pl"Pg"4-P2"P/-fP3"P4" = 0. 
Next, we have two planes U and U", and we therefore have a line which 
is the intersection of those planes ; when its coordinates are taken to be 
Pj'", . . ., Pg'", we easily find 
UiX,-l-U2X2-f-U3X3-fU,X,. 
The manner in which these inferred variables are used in the sequel will 
show that it is not necessary to retain the line P^'", . . ., Pg"'. 
Thus the sets of variables to be retained for the expression of con- 
comitants of the quaternary form are 
X, U, P, X", U", P". 
