MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
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whence 
x y' z! = — 4'670 = r\ 
[4.] .’. x' — x — 0*1 
y' — y — — 0-1709 r' — r — 0 - 1581. 
z' — z = 0-0709 
Now we can, without difficulty, calculate the approximate value of the arc (P P' in 
fig. 2.) subtended by the portion of abscissa a? — x (X X' in the figure). Calling this 
the arc (<r), we have 
Arc (x) =0-1351 
Arc (y) = 0-1817 
Arc ( 2 ) = 0-0715. 
And according to the theorem we ought to have 
Sum of arcs = f (r 1 — r) = 0-118. 
But in this example, arc ( x ) is to be accounted negative. Therefore we have 
Arc (y) + Arc (z) = 0-253 
— Arc (x) = 0*135 
Sum = 0-118 
which is in accordance with the theorem. 
Second example. — Suppose, as before, 
x = 1 
y = 1-7535 
z = —2-7535, 
whence 
xy z = —4-8281. 
And also 
x 1 — 2 
y = 0-8875 
z' = -2-8875, 
whence 
x'y'z' = —5-1253; 
both of which systems of values satisfy the given equations of condition. 
[5.] .-. x' — x = 1 
y — y — — 0"866 v 3 — r = — 0"297- 
2 ' — 2 = —0-134 
By calculation we find 
Arc (x) = 1-1319 
Arc {y) = 1-0443 
Arc ( 2 ) = 0-1350. 
But in this example both arc (^) and arc ( 2 ) are negative. Therefore we have 
2 b 2 
