MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
191 
simplicity into the subject, besides facilitating in no ordinary degree the progress of 
research. For instance, suppose there are 3 variables, and let x -f- y + z = p, x y 
-\-xz-\-yz — q and x y z — r, then x, y, z are the roots of the equation 
u 3 — p u 2 -f- q u — r = 0, 
where the variable u denotes indifferently either of the variables x, y, or z. This new 
letter u is only introduced for the sake of clearness, since we may equally well say 
that x, y, z are the roots of the equation 
x 3 — px 2 -\-qx — r = 0, 
or of 
y 3 — p y 1 + q y — r = 0, &c. 
This latter mode of expression is often more convenient. Now the function <p x, 
which we wish to determine. 
— y z = 
x y z r 
x x ’ 
and since p, q are here supposed to be given functions of r, we may find the value 
of ~ in terms of x, provided we can solve the algebraic equation 
a? — p x 2 q x — r = 0, 
with respect to r. 
Example. Let us resume the question concerning the sum of 3 arcs in the equi- 
lateral hyperbola. The equations of condition were 
x + y + z = 0, 
— x^y 1 z 2 
xy + xz +yz — , 
or 
— r 2 
x, y, z are the roots of 
r 2 
x 3 — x — r = 0. 
This equation (arranged according to the powers of r) is 
or 
whence 
and 
— x 
. — r + x 3 = 0, 
r 2 + — . r — 4 x 2 = 0, 
— 2 , / r 
r - ~ + 2 J * 2 + 
r — 2 o \f 1 + x 4 
V — ^ * x > 
which agrees with the former result. But now we are able to point out with clear- 
ness the limits of the possibility of the theorem. For the cubic equation 
■x 3 — - . a? — r = 0 
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