50 Colorado College Studies. 



conversely that by properly choosing the value of k, any conic 

 tliroagh these four points may be thus represented. Any of 

 the three conies may be a pair of lines. In the present case, if 

 P = and ^ = are our equations (1) and (2), it must be 

 possible to give^• such a value that the equation P + kQ = 

 shall represent the pair of lines, one of which is the line whose 

 equation we seek, while the other is the line joining the remain- 

 ing intersections of the given loci. But these two remaining 

 intersections are the two consecutive points on the circle in 

 which it is met by the two lines at c, s; accordingly the line 

 joining these points is simply the tangent to the circle at the 

 point c, s; that is, its equation is 



ex -\- sy = R-. 



And if we represent the line whose equation we seek under 



the form 



Ix + my -\- n = 0, 



(where Z, m, and n, are quantities at present undetermined,) then, 

 the form 



{ex -\- sy — R-) {Ix -\- my + n) = 0, 



or, as this becomes on expansion, 



cZa-^ 4- (Zs + cm) xy + msy"' + {en — eH — Is^) x 

 + {ns — c'm — ms') y — {e~n — ns') = 



is a form with which the equation 



Ax' + 2Hxy + P^- — 2 {Ac -{- Hs) x — 2 {Bs + He) y 

 + {Ac' + 2Hcs + Bs') + k {x' + y- — c" — s") = 



must become identical by giving a suitable value to k. 



Assuming this identity we have for the determination of 

 four unknown quantities, viz., k, I, m, and n, the six equations 



el = A -\- k, Is + cm = 2H, ms = B -{- k, 

 en — cH — Zs" = — 2Ac — 2IIs, 

 ns — C'm — ms" = — 2Ps — 2Hc, 



— c'n — ns- = Ac- + 2Hes + Bs- — c-k — ks\ 



The solution of the first four of these equations readily gives. 



