MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
5 
The roots of the equation then have the following values : 
x = 05 = sin 30° = sin & 
y— 0-94565 = sin 71° 1' = sin 8 
z = — 0-98154 = sin — (78° 59') = sin 8' ; 
and the theorem gives the sum of the arcs, or S 6 = const. The word sum is used 
in an algebraic sense, as including the case where one or more of the arcs are to be 
taken negatively , or its definition is 
S 6 = ± 6 ± 8 + 8\ 
The same ambiguity in the signs pervades the whole of this class of formulae. In 
the present instance 
S 6 = 0 + 8 — 8' 
= 30° + 7 1 ° T + 78° 59' = 180° 
the constant is a semicircle. 
Ex. 2. Let r = 0. 
x 3 — x = 0 ; 
and the roots are 
x = 0 — sin 0° 
y — 1 = sin 90° 
z = — 1 = sin — 90° 
6 = 0° 8 — 90° 8' ~ - 90° ; 
and the same formula gives, as before, 
6 + 8 - 8' = 180°. 
A very extensive class of formulae respecting the arcs of the circle may be obtained 
in a similar manner, by applying the method more generally. Thus, if we make the 
supposition 
C 1 -] = «o + «i x + + a n _ x x n ~\ 
where a 0 , a v a n _ 1 are constants, or any entire rational functions whatever of 
the variable v, we have an equation of 2 n dimensions, of which x is a root. 
If x — sin 0 t , and the other roots are sin sin 6- s , sin 0 2n , then 
/ dx 
and the other integrals = . . . 0 2n . And by a direct process we obtain the final 
equation 
S 6, or 6 l3 + 6 2 + .... + $ 2n =/• v + const., 
f . v being an entire rational function of v. 
But since it is generally admitted that no combination of circular arcs can be equal 
to an algebraic quantity, I conclude that we have generally 
/• v = 0. 
