MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
213 
§ 7- Example of an arithmetical calculation of the sum of three Integrals. 
The preceding analysis shows that the sum of the three integrals 
fyw - =? +fyr = r = const - 
if x^, y T i z* are roots of the equation 
t 3 — t — v = 0. 
But the form of this equation shows that the sum of its roots = 0, the sum of the 
products of every two roots = — 1, while the product of all the roots is a variable 
quantity = v ; the quantities x, y, z must satisfy the two following equations, 
x^ + f* + = 0 
{xyf + (x zf + {y zf = - 1. 
And if they do so, we shall have 
/+/,+/= const., 
denoting by^the integral 
But in order to eliminate the constant, we may take three other variables x?,y',z', 
satisfying the same two equations of condition, and thence deduce 
+X +i/' = const. 
Whence by subtraction we eliminate the constant 
(/.-/.) + {f-f) + (/-/) = o [»■] 
Now by the usual methods we find that the equations of condition are satisfied by 
the values 
x — *352342 
y = *917532 
* = 1*057860, 
and also by the values 
x' = *392456 
1/' = *900227 
z' = 1*065 602*. 
'.These values give 
** = — 0-209149 
y 4 — _ 0-878885 
1-088034 
x't = — 0-245862 
y '4 = - 0-854138 
2*4 — 1-100000 
Sum — 0 Sum = 0 
I his verifies the first equation of condition. The squares of these quantities are 
