192 
MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
being - compared with the form 
tf 3 — a x + b = 0, 
Jj2- q 3 y6 yA 
must have impossible roots when or when — > q~ ( -> or 1 > ^ x ^ or 
10 *27 > > 4 . Hence it appears that there are impossible roots whenever r is less than 
y 16 X 27 = 2 4^2 7 = ± 4*559. 
Accordingly in our numerical examples it will be seen that the values* of r are not 
contained within these limits. 
Another example. We have found (page 183. Ex. 3.) that the suppositions S# = 0, 
8 .r 4 = 2 r, give 
2 <p x = 2 x 2 + x + 4 oc 3 + x 2 . 
Here x, y , z are roots of x z + q x — r = 0, and we easily find from the doctrine of 
equations that 
S x 4 = 2 q 2 , .*. 2 q 2 = 2 r, .’. q 2 = r. 
Therefore x q x — q 2 = 0. Solving this quadratic equation with respect to q 
we have 
q — + Y x2 + 4 x 3 ’ 
But since 
7 
x 3 + q x — r = 0, — = x 2 + q- 
•*. ~ = 2 x 1 A x + ^/x 2 -f- 4 x? = 2 <p x, 
which agrees with the former result. 
Another example. Let $ -\- q x — r =■ 0, which gives for the first condition 
S x = 0. 
And let the second condition be 
r 2 + cr = q 1 + a q + h, 
a , b, c being constants. We find 
r * + g--*- a- + 
X 1 — X 2 ’ 
where X is a polynomial of 6 dimensions. 
Now let the n variables x, y, z be roots of x n — px n ~ 1 + i r = 0, 
where I continue to denote the product of all the roots by r; we have still <p x = — . 
Let the coefficients p , &c. &c. be replaced by their values in terms of r (which are 
supposed given, by means of n — 1 equations of condition). Then let the equation be 
arranged according to the powers of r, and the solution of it will give the value of r 
V 
in terms of x, and therefore the value of — = cp x. 
* These were r = — 4'83, r — — 4'G7, r — — 5' 13. 
