OF THE SECOND DEOBES. 



43 



(a.) When the line OQ3 meets the curve (1) in only one 

 point, (Fig. 6.) it is evident (v.) that APg will meet it only 

 in one point. In this case we shall have (i.) 



— ¥'\/{m] + 2m^cosy\-\) — Fv/(mj + 2wjCOSy+l) 



m, being the direction index of OQ.3 or AP3. Hence if we 

 assume 



D'»j,+E'=Dm,+E (b), 



we shall have F.A.P3=F.'OQ3 ; and by comparing this with 

 equation (a) we get 



AP,.AP, _ OQ,.OQ^ 



AP3 ~ OQ3 



(15). 



Substituting for D' and E' their values given in No. i., the 

 condition (b) becomes 



{Am^+B)t/,+(Bm, + C)x^=:o...: (V), 



and this must be combined with the equation 



Aw2 +2Bwii + C = o, 

 which expresses the condition that OQ.3 should meet the 

 curve in only one point (No. i). From the latter equation 

 we obtain 



— B+^/(B2— AC) — BHt^B' 

 Wj = — = — suppose, 



and by substituting this in equation (b'), we get, after slight 

 reductions 



B'{Ay, +(B+BOa?J=o (c). 



Now, when B'=o this equation is satisfied independently of 

 Xi and ^j, and therefore when the curve (1) is a parabola the 

 equation (15) holds good for every possible position of the point 

 A; but when the curve (1) is a hyperbola, B'is finite, and the 

 equation (c) cannot be satisfied unless x^ and y^ fulfil the 

 condition. 



Ay,+(B+B')^,=o. 



Hence we see that, in this case, the relation (15) will not hold 



