1889—90.] Mr R. E. Allardice on Solution of Equations. 139 
Notes on the Solution of certain Equations. By R. E. 
Allardice, M.A. 
(Read April 7, 1890.) 
The equations considered are equations of the third and fourth 
degree in a single variable, and systems consisting of two equations 
in two variables. This is a comparatively small class ; but if there 
be added systems consisting of a single equation of the third or 
fourth degree and a number of others all of the first degree, it will 
include all equations which admit, in general, of an algebraic 
solution. 
The object of these notes is to point out the advantage of making 
greater use, than is generally the custom, of the discriminant in the 
solution of equations, and to emphasise the importance of looking 
for geometrical illustrations of analytical methods whenever this is 
possible. The usefulness of such illustrations is well known in 
considering, for example, limiting cases in the solution of equations, 
such as the cases of infinite roots and of equal roots, of the meaning 
of which it is almost impossible to form an adequate conception, 
without the use of such illustrations. 
Consider the system 
U E ax 2 + 2 Inxy + by 2 + 2 gx + 2/y + c = 0 1 
V E a' x 2 4- 2 h'xy + b'y 2 + 2 g'x + 2 \fy + c' = 0 J 
This system may be solved algebraically by eliminating one of 
the variables, say y, and obtaining a biquadratic in x. This biquad- 
ratic may in particular cases be soluble by means of quadratics ; 
but, as a rule, it will require the general method of solution, by 
means of the reducing cubic. It will, however, in almost every 
case be found simpler to determine k so that U + kV shall resolve 
into factors. The equation in k will be of the third degree. If the 
solution of the system (A) can be made in any way to depend on 
quadratics alone, the cubic in k must have at least one rational root, 
which can easily be obtained in every case ; whereas the biquadratic 
in x will in general have no rational root at all. In the general 
case, when the solution of the system (A) cannot be made to depend 
on quadratics alone, the use of the cubic in k will save the trouble 
