22 
Proceedings of the Royal Society of Edinburgh. [Sess. 
the same elements to get those elements — not already got ; for there must 
be no reduplication of elements. So for the other lines. 
The triangular array thus created is seen to be the corner of an infinite 
rectangle. In other words, the solution of the quadratic being an expression 
singly infinite, the solution of the cubic is an expression doubly infinite, 
and we may so far anticipate by saying that the solution of the quartic, 
with all its terms present, is a triply infinite expression, that of the 
quintic quadruply infinite, and generally any solution of an equation of 
the n th degree, in which no terms are wanting, is an expression (n — l) ly 
infinite. 
Geometrically, the solution of the quadratic may be represented as a 
line, or an infinite series of elements ranged rectilinearly ; the solution 
of the cubic may then be represented as a surface, with such elements 
disposed regularly all over it ; the solution of the quartic is then an 
infinite rectangular parallelepiped or solid with such similar disposition 
of elements throughout it, but for the geometrical representation of 
solutions belonging to equations containing the fifth or higher powers 
of the unknown, and at the same time all lower powers, we require space 
of higher dimensions than 3. 
Returning to our solution, we remark that the left-hand side is the 
solution of the quadratic Rr 2 + Cx + E = 0. The right-hand side is the 
solution of the cubic Ax 3 + Cx -f E — 0. 
All parallel lines in this development, whether vertical, horizontal, or 
slanting, are the work of the same operator. The vertical ray, e.g ., 
E 5ABE 4 
C’ “c® - ’ 
- 180 
A 2 B 2 E 7 
etc., 
is, with all other verticals, the work of the operator Dp . Dp . Dp . D c 5 . 
The properties noticeable, in the indices and coefficients, which, of 
course, are constants, will not escape the reader. 
(h) Meanwhile we write down another solution, the one belonging to 
A,3 = -IW-C-E or *= -H3HSP 
On writing 
r= _ E + C/A\ E/Ay 
A + A\B/ AVB/ ’ 
we see the operators to be L);:' . Dj 1 , . D B 2 , and D,., 1 . Dy . D B 3 . The form ~ 
occurs at the outset with both. 
