ON THE HEDUCTION OF THE OENEE.AL EQUATION. 291 



Again, observing that the semi-sum of the values of r above found 

 is the distance between the focus and centre, and that this semi-sum 



is 1 s - (h cos 6 + k sin 6— p) and therefore = Ae, by (11), w© 



may write for the co-ordinates of the centre, 



h + Ae. cos 6, Jo + Ae sin 6 : 

 and the equation to the ' conjugate' axis becomes 



(x — h — Ae cos 0) cos 6 + (y — h — Ae sin 6) sin 6 — 0. 

 To find the points where the curve is cut by this axis, we combine 

 this equation, or 



x — h — At cos y — k — At sin 



siu - cos ~ r ' 



with the equation to the curve, 



(x — hf + (// — Tef = €2 (x cos 6 + y sin # — p) z : 

 substituting for x, y in terms of r, we obtain 



(rsin 9 + Ae cos 6f + (— ?* cos 6 + Ae sin 6) 2 



= e a (^ € + Jl cos # + fcain — p) 2 = £ 2 (ile H — A)\ 



or, 



r 2 -M 2 e 2 =^ 2 . 

 giving two points, which are real in the ellipse, and imaginary in the 

 hyperbola. Hence denoting the intercepted part of the conjugate 

 axis by 2B, we have 

 B*=A 2 (1-e 2 ) 



We may now go on to discuss the varieties of form which the curve 

 may assume for particular relations among the constants. 



I. In the elliptic class, where ab— e 2 is positive. 

 Here m is always to be taken with the positive sign, and (a + b + m) , 

 and (a + b—m) are both finite and positive, and A and B are there- 

 fore either both real or both imaginary ; also they may vanish toge- 

 ther, but neither of them can become infinite except by passing into 

 the parabolic class. 



Also ab being greater than c 2 , ae 2 + bd 2 — 2cdo is always positive, 

 and therefore if /' be negative, the curve is always real : if f be posi- 

 tive, the curve is real or wholly imaginary according as ae- + bd 2 — 

 2cde is greater or less than (ab—c 2 )f. 



If ae? + bd 2 —2cde=(ab—c 2 )f, then A and B both vanish, and tho 

 curve is reduced to a point whose co-ordinates are given by (10) and 



• The value of Ii might have been deduced from that of A by changing the sign of m 



also K might havo been deduced from 11 by changing the sign of m, and writing ( ZL+0) 



2 

 for 6. This might havo been inferred from consideration of tho imaginary directrices 



