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MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
Ex. 2. Let x — — y = — be assumed for two roots of the equation ; in which 
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case we find a — — g-, b = y, and the third root z — — 
Since xy — 1 in this example, the theorem gives the sum of three arcs = xy z — z, 
which we propose to verify. 
Now the formula 
2 arc x = x \/\ + X 2 + log 0 + \/l + x 2 ) 
gives 
2 are ( 4 ) = || + l°g 2 
2 arc (-j) = - + log3 
the sum of which two = — + log 6 
and 2 arc + lo g 6. 
Therefore the sum ( subtractive ) 
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arc x + arc y — arc a = ~ 
But on the other hand we have 
Therefore the sum of the arcs = z i which was to be shown. 
§ 4. Analogous Properties of the Circle and Parabola . 
There is a manifest analogy between the area of the circle and the arc of the para» 
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bola, the former being expressed hy^fd x V 7 ! — x 2 , the latter byj dx^fx -J-x 2 , which 
only differ in the sign. The same analogy is seen in the theorems which may be de- 
duced respecting these integrals. Thus, for instance, the Theorem II., which we 
have demonstrated in the parabola, may be applied, with a slight modification, to 
the circle. If we put 
v \/ 1 Hh x 2 — n x 2 + x + v, 
we find the sum of three integrals of the form 
J* d x \/l-Px 2 — 4 ~ r, 
the constant being = 0. The upper sign applies to the parabola, the lower to the 
circle. The demonstration of the latter case is omitted for brevity, being exactly 
similar to that of the former. 
