288 ON TflE REDUCTION OE THE GENERAL EQUATION. 



In this expression for the excentricity, m may bear either sign ( + 

 or — ), but we observe that wheD, as in the ellipse, ab — c 2 is positive, 

 which requires a and b to have the same sign, and therefore (since a 

 is essentially positive) a-\-b to be positive, m is less than a + b, and 

 the negative value of on makes e impossible. So also in the parabola, 

 where ab — c 2 =0, the positive value of on gives 6=1, while the nega- 

 tive value makes e infinite. 



Hence in the ellipse and parabola, the positive value of m must be 

 taken: but in the hyperbola, where ab — c 3 is negative, either sign 

 gives possible values to e, one of them referring (as will afterwards 

 appear) to the hyperbola, and the other to its conjugate, and the two 

 values are evidentlv connected by the relation 



El* + 6 2 * 



It will be shewn in the sequel how to discriminate between them. 

 We have now 



m a+b + m ' 



and substituting for A and e a in (1), (2), (3), we find 



~ •' 2 2 \a , a+b 2 a 



2 cos 2 6 = = 1 + — — — 









e« 



e- 







= 



1- 



a—b 

 in 



2 



sin' 



■6 = 



.1 



a-b 

 1 



T 



m 



sin 



COS 



= 



c 



Of the four values for 6 determined by either of the first two equa- 

 tions (m bearing a determinate sign) the third equation will shew 

 which of the pairs, namely, and tt + 6, or tt — 6 and 2-—0, is to 

 be selected, and it is then indifferent which of the angles in that pair 

 we take, due regard being had to the direction in which p is to be 

 drawn from the origin as indicated by its sign ; for, the change of 

 into tt + 6 in our original equation only changes the sign of p, and we 

 thus obtain in both cases the same determinate position for the 

 directrix. 



There remain now the equations (4), (5), (6), from which to com- 

 plete the determination by finding p, h, and k. Eliminating h and 1c 

 from these equations, we have 



p" + f- — (d cos + e sin0) : — -— - — f = 



Prom this we perceive th t there are two and only two directrices 

 corresponding to these t.vo values of p, (for is restricted to one of 



