An introduction to the modern theory of equations, by Florian Cajori.

ELEMENTARY TRANSFORMATIONS OF EQUATIONS 39

Let the roots of equation II be a,,, y. Then the squares of
the differences of every two of the roots are
( -/P)2, ( -y )2, (- y)2. III
Since the roots of TI are the roots of I, each increased by b-, it
follows that the dicferences of the roots, two by two, of equation II are the same as the differences of the roots of equation I.
Hence the squares of the differences, given in III, are the
squares of the differences of the roots of equation I, as well as
of equation II. In other words, both equations lead to the
same "equation of squared differences." This last equation is
evidently
z _ -- (a - - 2} -( )- ( ( y) = O. IV
The coefficients may be calculated as follows: Equation IV is
satisfied by the equality
z= (_- )2.
We obtain from this
z = e2 +3- 9 + Y2 2 _ 2 _y.
Now ta2+-/2+7Y2 was shown in ~ 15, Ex. 2, to be equal to a —2 a2;
3H
in the case of equation II, a, = 0, a = * So,
b02
C2 + 9 + 72 6 =
0
while af/y = - -
Hence we may write
6 H 2 G
Z = -.- y2 +
where y2 and y are written for 72 and y. This is allowable,
since y is one of the three possible values that y can assume in
equation II.