186 
MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
fore a corresponding- change to take place with respect to y and z (since they are 
functions of x). These new values may be called y' and z'. 
At the same time the product xy z = r is changed to x' y 1 z' = r ' ; and the first 
set of arcs, which may be denoted by A, are changed for a second set, which may be 
denoted by 13. 
Now the original equation gives, sum of arcs A = f r + C, and the changed equa- 
tion gives, sum of arcs 13 = f r' + C ; .*. by subtraction, sum of arcs A — sum of 
arcs 13 = f (r — r'), in which result the constant is eliminated. 
The accompanying figure 1. represents two opposite equi- 
lateral hyperbolas, with their asymptotes. C is the centre, 
and origin of the abscissae C X = x, C Y= y, C Z = z, of which 
the latter must be negative (supposing the two former posi- 
tive), by reason of the equation x + y + 2 = 0. Therefore 
it belongs to the opposite hyperbola. If X P is the ordinate 
corresponding to the abscissa C X, the equation of the curve 
is C X . X P = 1, and the arc subtended by C X is the infinite 
arc O P. 
Fig. 2. represents the abscissa C X = x, both in its original and in its altered state 
when it has become C X' = x'. In the former case it subtends the Fig. 2. 
infinite arc O P, and in the latter case the infinite arc O P'. But 
in taking the difference there remains the portion of abscissa 
X X' subtended by the arc P P', which is a finite quantity, and 
thus the embarrassing consideration of the infinite arcs is avoided. 
Now the sum of arcs A — sum of arcs B = sum of three limited 
arcs, of which P P' is one, and the others subtend the portions of 
abscissae Y Y' and Z Z'. Denoting these arcs by K, we have this 
equation in finite terms : 
Sum of arcs K = | (j — r). 
Now in order to put this equation to the test of numerical computation, it is requi- 
site to find three quantities that verify the equations 
— r 2 
Sx = 0 S x y — — ~ — 
Suppose, therefore, 
x = 1 
Fig. 1. 
y = 17535 
z = — 2*7535, 
whence 
xy z = —4*8281 = r. 
The equations are satisfied by these values, and also by the following : 
x> — 1*1 
?/ = 1*5826 
z' = —2*6826, 
