THEIR APPLICATION TO THE SOLUTION OF EQUATIONS. 177 
quently, when n=5, 7,_,(v) cannot in general be rendered 
symmetric. 
6. To show the bearing of this theory on the solution 
of equations. Let 2,, 2%, %3,-++x, represent the roots of 
an equation of the 2 th degree, 
xv" +ax" + ba” +--+» +sx+t=0, 
and consider the effect of supposing 7r,,_;(~) to vanish. 
For the quadratic 
xv +ar+b=0, 
we have (Art. 2) 
(x) = V (ay +2Xq)*— 4a, = V a — 4b, 
and the evanescence of 77,(z) gives b= ja’, 
*, @+axr+ia=0, or (#+4a)’?=0, 
which is immediately solvible. 
7. For the cubic 
a+ azxr’+br+c=0, 
we have (Art. 3) 
T(x) = (Sx)? - 38a, %=a — 3b, 
and the evanescence of 7,(x) gives b=4a’, 
“. P+axr+4+ie¢ x+c=0, 
or (w+4a)*— (3a -c)=0, 
which admits of easy solution. 
8. For the biquadratic 
a+az*+ ba’+cr+d=0, 
we have (Art. 4) 
T(x) = (Lx)> — 42a La, 2+ Bra, #2 Xs 
= -a>+4ab-8e, 
and the evanescence of 7;(”) gives 
b= i@ + cs 
a 
*, @+ae+ (3084 )a* + 00+ a=. 
a 
or («+ 4ar-£) - (4-4) =0, 
which also admits of being easily resolved. 
