MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
189 
-fZ'ftT*, 
which justifies the assertion, that in this example arcs ( x ) and (z) are to be accounted 
negative, and arc (y) positive. 
Perhaps this reasoning may not be altogether free from objection : I wish it, there- 
fore, to be remembered that I am not giving it here as being the most convenient 
method of determining the signs of the arcs, but merely as being the reasoning I 
employed at the time * when I first met with this theorem. 
This theorem shows that three hyperbolic arcs may be determined in an infinity of 
ways, so that their sum may be an algebraic quantity. At the same time it shows 
that one of these arcs cannot be supposed always to be 0, so that Fagnani’s theorem 
respecting the sum of two arcs is not an instance or particular case of this. I have 
dwelt at some length on this theorem, because the theory of the conic sections has 
always been regarded as so important by mathematicians that any considerable addi- 
tion to it is thought deserving of attention. 
I now proceed to other results which presented themselves in the course of this 
inquiry. 
Still continuing to suppose the variables to be three in number, it is allowable to 
suppose between them any two symmetrical equations whatever ; and thence if we 
can deduce the value of yz or tp x in terms of x alone, we may apply the general 
theorem A. 
Ex. 1. Let Sx = a, and S xy = ( — ) , we find 
yz_ L \/l — x 4 + ax 3 
2 x' 2 ' x 2 
Ex. 2. Let S x = a, and S xy — \/ 2 b .xy z, we find 
y z =. x 1 — a — b .x -f- \/ 2 b x 3 -f- ( b 2 — 2 a 6) x 2 . 
Ex. 3. Let \/x + */ y -f- */z = \/ a (or S\/x = V a), and let S# = b, we find 
, a — b r 
V yz = ~ 2 V ax -\-x —fx, 
whence 
y% - <p x = [fx] 2 . 
A great variety of different suppositions of this sort may be made ; but if the re- 
sulting function <p x should become too complicated, little practical advantage would 
be derived from the knowledge of its properties. I therefore thought of another 
method of obtaining this function, by means of what may be termed “ changing the 
conditions.” Thus let the original equations of condition be 
x + y + z = 0, and xy x z y z — — 1, 
* 1825 . 
