ON THE REDUCTION OF THE GENERAL EQUATION. 287 



will be denoted by m, can always be made identical with the equation 



(x—h) 2 + Q/—Jc) 2 =z 2 (xco86+g sin<9— p) z : 

 for this latter is an equation of the second degree with all its terms 

 complete, and containing the requisite number of arbitrary constants. 



Since the left-hand member oi this equation is the square of the dis- 

 tance between the points (x, y) and (h, k), e is a constant, and the 

 other factor of the right-hand member is the square of the distance 

 of the point (Ji, k) from the line x cos 6 + y sin — p=0, it follows 

 that the general equation of the second degree expresses the locus of 

 a point whose distance from a fixed point (real or imaginary) is 

 always proportional to its distance from a fixed, real or imaginaiy, 

 straight line. 



Adopting the usual nomenclature, the point (Ji, 7c) is a, foe us, e is 

 the excentricity , and the fixed line x cos 6 + t/ sin 6 — p = is a 

 directrix. 



Multiplying the first equation by an arbitrary quantity ( X ) ; 

 arranging the second equation by powers of the variables, and then 

 equating corresponding coefficients, we obtain the 3ix following equa- 

 tions from which to determine the six unknowns, c, h, k, X, 0, p ; 



Xa — 1 - e 2 cos 2 6 (1) 



Xb = 1 — e 2 sin 2 6 (2) 



Ac =— e 2 sin0cos0 (3) 



— \d = h —p e 2 cos 6 (4) 



— Xe = Jc — pe 2 sin 6 (5) 



Xf = k 2 + £ 2 — ^ 2 £ 2 (6) 



By taking (1) x (2) — (3) 2 , we obtain 



A 2 (ab— c 2 ) = 1 — e 2 (7) 



Hence, according as ab—c" 1 is positive, zero, or negative, c is less 

 than, equal to, or greater than 1, corresponding respectively to the 

 three varieties of the ellipse, parabola, and hyperbola. 



Also from (1) + (2) we obtain 



X {a + b) = 2 - e 2 (8) 



and then (8) 2 — (7) x 4 gives 



A 2 | (a + b) 2 — 4 (ab— c 2 ) I = e* 



or, substituting m, 



X m = e 2 



from which, by substitution in (8), we have 



2 m 



c 2 = j (9) 



a + b + m v J 



