168 MR. MURPHY ON THE ANALYSIS OF THE ROOTS OF EQUATIONS. 
When the quantity y) in the expression for the root which is unaffected by 
surds vanishes, one root is equal to the sum of the other two with changed signs. 
When — y and a" vanish conjointly, two of the roots are each one half of the third 
with a changed sign. 
When — y and a! vanish jointly, the equation has two roots equal, but with con- 
trary signs, and the third root is zero. 
When — y, a', a” all vanish simultaneously, the three roots are equal to each other 
and to zero. 
24. Biquadratic Equations . — Let x X) x 2 , x 3 , x 4 be the four roots of the equation 
a? 4 -j- a x 3 -f- b x 1 -j- c x -j- d, 
the number 4 being composite allows the subdivision of the sums of the roots taken 
two and two into three pairs, which have rational sums, thus 
+ x 2 + y ~ ~ ( X 3 + x i + y) 3 (#1 4- x 3 + y ) = — (^2 + ^4 + y)> 
x i + x a 4- y = - (* 2 4- % + y), 
therefore the equation, of which the roots are 
x i 4 x 2 + Xl + 45 + y> x i 4 x ± 4 y, — { x 3 4- x 4 4- y), &c. 
is one of six dimensions, but without terms involving the odd powers of the unknown 
quantity, and therefore these quantities are the square roots of the roots of a cubic 
equation. Put therefore 
x i + ^2 4 I" = 2 4 a 
4i 4" x 3 4- y = 2 4 (3 
+ x 4 4 y = 2 4 y, 
therefore 
3 CL 
2 X { 4- (#1 4- x 2 4 x 3 4 a? 4 ) 4- -y = 2 ( 4 a 4- + 4 y), 
‘£i= — y4 4a-}-4/£>4'4y, 
the quantities a, |3, y being the roots of a cubic equation, are of the form 
a — a 1 4 " J ft 4 \/ ft' 
j8 = a' 4-^\/(3' + 
y = a' + ^2^/(3' 4 (3", 
