OF THE SECOND DEGREE. 



39 



for the direction-index of the ordinates to the axis. Substi- 

 tuting this value of m in equation (10), we obtain, after slight 

 reductions, 



{(A-fC) cos y— 2 B}(B3/ + Ca:)-|-(B D-l-C E) cos y 



— (C D + B E)=o (11), 



which is the equation to the axis of the curve (1) when equa- 

 tion (1) represents a non-central curve, 



(/9.) In the case of a central curve, since m' is the direction 

 index of the straight line (4), we have 



A m m' + B (m -j- w') + C = o ; 

 hence, by eliminating m by means of equation (a), we get 



(A cos y— B) m' H(A— C) «i'=C cos y— B (b.) 



Solving this by the common rule for quadratics, we have 



^ C— A ± V { (A— CP+ 4 (B— C cos y) (B— A cos y)} 

 '^- 2 (A cos y— B.) 



Now, since the suffix of the radical in this equation may be 

 written in the form, 

 (A.—Cf (sin2 y 4- cos^ y) -|- 4 B2— 4 B (A-j- C) cos y+4 AC cos^ y, 



or {(A— C)siny}^+ {2B— (A H- C)cosy}^ 

 it follows that the roots of equation (b) are always real; and 

 thus we see that every central curve of the second degree has 

 two axes, and cannot have more than two. 



IV. 



Returning to the general formula in No. II, let the point A 

 coincide with the centre C of the curve (1), the straight line (2) 

 cutting the curve in P and Q (fig. 3); then by equation (3) we 

 shall have 



CP cQ- ^"^'^'+^'^^°'y+^^ J 

 A r»2 -f 2 B r» + C 



where F'= Ky.,^+ 2 B 2/2 3:3+ C 0-2' + 2 D 2^2 H- 2 E xj-i- F, 



=:D2/2+Ea:2+F (by No. II), 

 and the values of x^ and y^ are given by equations (9). But 

 since PQ is bisected in C, we have CQ,= — CP, and the pre- 

 ceding equation gives 



