1889-90.] Mr K. E. Allardice on Solution of Equations. 141 
which may be transformed into the above reducing cubic by 
dividing the roots by q 2 and then forming the reciprocal equation.] 
As a last illustration consider the system * 
x 2 + y = a, y 2 + x = b . 
The resultant in x is 
set- 2ax 2 - s tx + a 2 - b = 0 ..... (1). 
Now in the general biquadratic (deprived of its second term) 
set +px 2 + qx + r = 0, put x = \y ; then 
f + (p/X 2 )y 2 + (q/X 3 )y + r/A 4 = 0 . 
Now put q/ A 3 = 1, that is A = q 118 , and the equation becomes 
y 4 +py 2 + y + t — 0, where p' —pj A 2 = pfq 218 , r = r/q* 18 . 
This may be identified with (1) by putting 
-2 a = p' and a 2 - & = / ; that is a = —p'/2 , b =p ' 2 / 4 - r'. 
Hence the general biquadratic may be solved graphically by 
means of the two parabolas x 2 + y = a and y 2 + x = b. 
The advantage of this method is that these two parabolas only 
vary as regards position, a change in a causing the first parabola 
to move along the y-axis, and a change in b the second parabola to 
move along the a?-axis. In applying this method in practice one of 
the parabolas might be drawn permanently on a sheet of paper 
ruled in squares ; and instead of shifting the parabola it would be 
sufficient to change the position of the origin of coordinates. The 
second parabola would require to be drawn on a sheet of tracing- 
paper or engraved on glass, f 
The above method fails when the roots are equal in pairs, for in 
this case ^ = 0. This is also evident geometrically, since two 
parabolas cannot have double contact unless their axes are parallel. 
Three roots of the given biquadratic will be equal when the two 
parabolas have contact of the second order. The point of contact 
will obviously lie on the line y — x, and the tangent at the point of 
* The idea of using this system of equations for obtaining a graphical 
solution of the general biquadratic is due to Professor Chrystal. 
t When this paper was read a diagram was shown illustrating the graphical 
solution of several biquadratics. 
