342] 
THREE GIVEN POINTS AND TOUCH A GIVEN LINE. 
263 
which shows what has been all along assumed, that the curve is of the fourth order. 
This equation may be transformed into 
x- (a 2 a? + /3-?/ + 7 V — 2/3yyz + 2y<azx + 2 a/3xy) 
+ y 2 (a 2 x 2 + ¡3-y- + rfz* -I- 2 ft y yz — 2yazx + 2aftxy) 
+ z 2 (cl 2 x 2 + j3 2 y 2 + 7 2 z 2 + 2ftyyz + 2yazx — 2 aftxy) 
- 2yz (ax + /3y + 7z) (— ax + fty + 7z) 
- 2zx (ax + fty + 7z) ( ax — fty + 7z) 
- 2xy (ax + fty + yz) ( ax + fiy - yz) = 0 : 
ii with this equation we combine the equation ax + /3y + yz = 0, we find at the points 
of intersection with the given line 
so that the points in question are the intersections of the given line ax + /3y + yz = 0, 
corresponds to the conic which touches the given line at its intersection with the 
assumed line x + y + z = 0, the pole in relation to this conic is obviously a point on 
the given line. The point in question, if x + y + z = 0 denote the line infinity, is the 
point I of fig. 3. 
It may be proper to mention a far less symmetrical form of the equation of the 
conic, but which has the advantage of putting in evidence the point of contact; viz. 
the equation is expressed in terms of the parameter a denoting the distance of the 
point of contact from a given point in the base line, and which is therefore very 
convenient for tracing the changes of form of the conic. Assuming as before that the 
base line cuts the sides produced, then (see fig. 4) if of the three points 1 denote 
Fig. 4. 
a' 
0 « 
x 
that which is furthest from, and 2 that which is nearest to the base line, and if 
the base line be taken as the axis of x, and 23 as the axis of y; the equation of 
the base line is y = 0, and the equations of the sides 23, 31, 12 are x = 0, 
- +1 = 1, where a, b, a', b', a - a, b' — b, a’b - ab’ are all positive, so that, by choosing 
a b