182 
Proceedings of the Royal Society of Edinburgh. [Sess. 
(which is a line-quantic), being a quadratic line-complex,* in the point- 
variables X, the line-variables P, and the plane-variables U, together with 
the variables X"', P", U" of § 11. 
27. The construction of a complete system of independent concomitants 
for the two quantics existing simultaneously is facilitated by a knowledge 
of the complete system of concomitants for each of the quantics taken 
separately. 
28. The complete system for the quantic 
2 = 1), c, ri,/, g, h, I, m, n ^ X 3 , XJ 2 
is easily deduced from known results. There are the fifteen partial differ- 
ential equations to be satisfied ; and the number of quantities they involve 
is twenty-four, viz. fourteen variables and ten constants. Thus the system 
must contain nine independent integrals. 
We write 
.a . ^ . d 
x/^-i-x/^ 
aXi'""ax.2 " dx^ 4 ax. 
0u=u;A+u/ ® +u/J^ + u/ ^ 
. a 
.a , ^ . d 
au/ 
p/^+P 
„ 8 
aPj ■ "■' 8P^ ' “ ePj 
+p,; 
■A 
A' 
Then in addition to 2 we have 
V 
~^1 
la 2 
~'2~ 2^ X ^ 
* The quadratic line-complex was first considered by Pliicker, “ New Geometry of 
Space,” Phil. Trans. (1865), pp. 725-791, and subsequently in his Neue Geometrie des 
Raumes (1868). 1 have preserved his notation so far as regards the coefficients of the 
complex, because it has been used by other writers, and variations of notation tend to be 
confusing ; but a notation which runs 
^12 9'i3~P 5 
?22 = P 5 ?23=P-5 
!?33 ~ T , ^34 = V , 5^35 ~ fi ? 
?46 ~ 1 
?14“T , = S 5 
5'24 — P 5 ^25“^? ?26=P 5 
^36 — Q? ?44~P ? 745 
755 = P > 7o6 J 5 7g6 ) 
would codify the expression of the operators. Thus the operator Hj becomes 
d7i6 a726 a736 ^746 ^756 ^766 
and so for the others : the relation N- 4 -O-f V = 0 becomes 
7i6 + 725 + 734 — 
and similarly for other invariants : the notation immediately suggests (or is suggested by) 
the umbral notation used in my memoir which has just been quoted. As the umbral 
notation is not used here, I have adhered to the Pliicker coefficients. 
