MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
9 
Therefore 
S*Jl+x 2 .dx = — 
S J* J\+x 2 .dx= — ^ = r. 
Now writing n — , v — b, we have the equation in the form given above, viz. 
x 9 — a x 2 + — a b — x + ~ = 0 ; 
and therefore the theorem is demonstrated. 
Examples. 
Ex. 1 . Let a = 2 + J2, b — 1 , the roots of the equation are 
x — 1 
i 5 1 
y ~ 2 ' - 1 
x — 1 .*. arc x — 1-147793 
y — 3-906278 .*. arc y — 8'91 1399 
2 = - 1-492065 .-. arc z = L935186 
and 
Now we have 
.-. S x = 3-414213 = 2 + </2 = a 
xy z — — (3 -{- 2 2) — — 5-828426. 
Arc y = 8"911399 = (1.) 
Arc x + Arc 2 = 3-082979 = (2.) 
Sum ( subtractive ) = — 5-828420 = (2.) — (1.) 
xy z = — 5*828426 
Error = 0-000006 
The quantity which we previously called f x — log {x -f- 1 -f # 2 ) has the follow- 
ing values : 
fx = 0-881372 
fy — 2-071728 
fz = 1-190354 
we have fx + fz = 2-071726 
fy = 2-071728 
Error = 0-000002 
Thus it is seen that the logarithmic parts destroy each other as in the first theorem. 
mdcccxxxvii. c 
