AND IMAG-INAEY BOOTS OF EQUATIONS. 
635 
Let J 2 =<£K, J 3 =*L. Then 
G J , _8 2 72 432. 
§ 4 vg 3 vq* v 2 g v 3 1 
J 9 
and 
J 2 Q 
We may for the moment make abstraction of the section of G made by the plane of L; 
that being done, and J, K, L being referred to the form nt 5 +sri+£M; 5 or nt*+sw®+#M> 5 , 
calling +°, M, and, as before, using g, <r, r to denote st, tr , rs, we have 
+ J =M(g 2 +<r 2 +r 2 — 2 g<r — 2gr — 2or), 
K = M 2 g<7r( § + <r + r), 
+ L=M 3 gW. 
Now when G=0, we may suppose 
We have then 
g=<r, T =l=0 + 4, 0 being a new auxiliary variable. 
? <r 
+ J=M(r 2 -4gr) =M f r0, 
K=MYr(2 s +r)=M’ f v(l+ # -| i ). 
+ L = M*g V = M 3 gV JL , 
and consequently 
* =^=0 4 +40 3 , 
Li 
/y J 2 5 2 (A+4) _ 
1 K 0 + 6 
(56) In general we have 5 4 -|-45 3 — v= 0. 
By a well-known corollary to Descartes’s rule this equation can never have more 
than two real roots ; when v is positive there will always be two real roots of opposite 
signs ; but when v is negative and inferior to a certain negative limit, all the roots become 
imaginary. When v lies between zero and that limit, two roots of 0 will be real and 
both negative. To find that limit we may make 40 3 +120 2 =O, or 0= — 3, which gives 
*=81 — 108=— 27. 
(57) When D=0, <£=-=128, i. e. 0 3 +40 2 -1280-768 = O, or (0+8) 2 (0-12) = O ; 
so that the roots of 0, when D=0, are — 8, — 8, 12, and the corresponding values of * 
are 2 U , 2 11 , 2 10 27. 
If now we make 0 4 + 40 3 = 2 u , one of the real values of 0 we know is — 8, and the 
other will be the real root of the cubic equation 0 3 — 40 2 +320 — 256 = 0. 
j When 0 = 5, the left-hand side of the equation =125+160 — 100 — 256= — 71. 
j When 0 = 6, the left-hand side of the equation =216+192 — 144—256= 8. 
Hence the real root lies between 5 and 6, and q lies between ^ and Thus 
<£<30 and ^=1 — ^ is negative. 
Again, if we take 0 4 +40 3 =27'2 10 , and take out the root 0=12, the resulting cubic 
becomes 
0 3 +160 2 + 192 0+23O4=O, 
j where it will easily be seen the real root lies between —12 and —16. 
mdccclxiv. 4 Q 
