114 
This is the equation for the determination of the points of intersection of L’ 
and the conic. One solution is known, viz.: x =x. Hence x — ita is a factor. . 
Divide by « — va and call the resulting equation: 
(7.) X (a, 4)=0. 
This equation is linear in x and therefore can be solved for x by rational pro- 
cesses. Let the solution be: 
(8.) “= (A). 
(9.) . y=o[y (A) }. 
(10.) dz = ’(A) da. 
(11.) “fF (x, y)de= [F { (4), 9 Lv (4) ] \w (a) ar. 
Since F, 0, /, ¥’ are all rational functions, it follows that 
F 4} ¥(4), 91(4)] bw). 
is a rational function of 2. Call this function r (4). 
(12.) ve (x, y) de =r (2) da. 
As an illustration corsider the integral : 
{F (x, y) dx, 
y=V ar? + ba-+ c?. 
(1.) y?=ar?+br+c?. 
Take A as the point (0, c) 
and as the Jines through A: 
(2.) Y=, 20: 
Then the equation of L is: 
(3.) y—e+Arx=o. 
(4.) -. Y= (4, A) =c— Ax, 
(5.) .*. y2 = (c—Az)?. 
(6.) (¢ — Ax)?— (ax?+ ba + ce?) = 0. 
(7,) .. X (2, 2) = (A* — a) e— (2¢ 4 + b)=0. 
: 2ch +-b 
(8.) ae e)) Sloe 
ae ay eVtd+ac 
(9.) y=o[ (4)]= RE fa 
— -_-. . 
