4 
MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
These theorems, and the analogous ones which exist between any number of tan- 
gents, are well known. But when we apply the method to the integral J x — , 
we obtain relations between circular arcs which appear to be of a more novel de- 
scription, and perhaps have not hitherto been noticed. Of which I will proceed to 
give an example. Let us suppose 
vrr7 = vx + l ’ 
whence 
x z + — x 2 + ( - 2 - — 1 ) x — = 0. 
' v \ v / v 
2 
In this instance the symmetrical v = — , and therefore making this substitution, 
we have 
x 3 
-J- r x 1 + — 1 ^jx — r = 0. 
There are therefore two necessary relations between the three roots, viz. 
And since 
But 
Also 
P = ~r q = -~ l. 
" 7 t = 5 = vx+1 
VI — X 
S 
d x 
Vl - 
— v$xdx-\->z>dx. 
S 
fr- \ r 2 
x 2 = //2 - 2 q = r 2 - - 2 ) = ^ + 2 
S x d x — 
r dr 
2 
2 rdr 
S x d x = — . — = d r. 
r 2 
S d x = — dr 
•. S 
V' 1 — 
■=. dr — dr — 0 
whence this theorem : 
7/ 1 the sines of three circular arcs are roots of the equation 
x 3 -f- r x 2 + — ] ) x — r = 0, 
the sum of the arcs is constant. 
I will give a numerical example of this theorem. 
The value of r is arbitrary. Suppose it to be 
= 3 — */\2 = — 0 - 4641016 . 
