MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
7 
S^ 3 — 8i? , „ , 9 , 2U 3 
— = —3 1 - 2 v (v 2 + 1 ) - 2 v = 3 -. 
To find the value of the second term J* v S x d x, we have 
S x 2 = p 2 — 2 q = 4 v 2 — (2 v 2 + 2) = 2 r 2 — 2 
.\Sxdx = 2vdv 
and 
2U 3 
f v & x d x = /' 2 v 2 d v — 3 . 
Therefore these two terms destroy each other. Consequently we have simply 
s/* ^^1 + ^ = Sx + c. 
It appears by trial that C = 0 , and the equation between the roots [ 1 .] becomes, 
— V 
by writing for v its value — - — , 
x 3 — r x 2 -\- -j- 1 ^ x — r = 0 
.*. the sum of three arcs = S x = r. Q.E.D. 
Example . — Let us suppose r = 4 + 2 */2 
= 6-82842 7. 
The three roots will be 
x = 1 
# = 4-2042580 
z = 1-6241690. 
Calculating the arcs accurately by the formula [ 2 .], we have 
Arc x = 1-147793 
Arc y — 10*156004 
Arc 2 = 2-179773 
In forming the sum we must notice that arc x and arc 2 are to be accounted nega- 
tive. Consequently we find by subtraction, 
Arc y = 10-156004 
Arc x + Arc 2 = 3*327566 
Sum = 6-828438 
r = 6-828427 
Error of calculation = 0-000011 
Thus the calculation verifies the theorem with considerable exactness, and shows 
that no constant is required to be added to the integral. 
Since the sum of these three arcs is algebraic, and that each contains a logarithmic 
part, the sum of these three logarithms must be = 0 : for if not, it must be an alge- 
