762 PEOFESSOE BOOLE OX THE COMPAEISOX OF TEAXSCEXDEXTS, TTITH 
give the value sought. If the number of integi’als under the sign 2 is specified, suppose 
it w, the function F must be so chosen that the reduced equation E=0 may be of the 
wth degree. 
The algebraic sign of each of the integrals in ^yK-dx will be the same as the sign of 
the corresponding function F, which, as being rational, is not ambiguous. 
17. Example 1. — The following theorem is given by a writer hr the Cambridge Mathe- 
matical Journal, vol. i. p. 268, as a generalization of a theorem of Mr. Fox Talbot's 
relating to the arcs of the equilateral hyperbola. The equation of any hyperbola referred 
d^ + b^ 
to its asymptotes being xy = — or for simphcity, xy=m^, we have, supposing d the 
angle betweerr the asymptotes, 
C cos + m"* 7 
aic _J -5 dx. 
Supposing then three values of x to be determined by the equation 
's/ x^ — 2m^x^ co^&-\-m^=vx-\-iifd, ( 1 .) 
the sum of the correspondirrg arcs will be 
3 , nd cos fl , . , . 
2 '^H — +const (2.) 
To demonstrate this theorem, it must be observed that the equation (1.), reduced to a 
rational and integral form, becomes 
x^—{2m^Q,od‘6-\-v^)x—2iifv=0, (3.) 
which occupies the place of E=0, art. 16. By vhtue of the same equation we have 
''vx -f- nd 
V" x'^ — 2rrdx^ cos 6 -f- 7rd 
dx 
dx. 
Thus we have, 1st, to transform the expression 
^vx + rrd ^ 
2 “2 dx, 
(^•) 
the simultaneous values of ^ being determined by (3.); 2ndly, to integrate the result with 
respect to the new variable v. 
Applying the general theorem of transformation, art. 12, we have 
vx + mr 
x^ 
dx=z e r -[2xv^2m^)lv 
L x^ — {2nd cos ^ + v^]x — 2ndv ^ ’ ' 
i + tjq 
Here we must develope the function 
vx-\-rfd 
x^ ^ x^ — (2m^ cosS — 2ndv ' 
— {^Ixv 2nd)tv 
( 6 .) 
in ascending powers of x, and take therein the coefficient of From this we must 
X 
subtract the coefficient of - in the development of the same function in descending 
powers of x. 
