OF THE SECOND DEGREE. 



45 



which determine the origin of the new co-ordinates introduced 

 in equation (16.) 



(y.) When equation (1) represents a parabola we are now 

 prepared to determine the elements which fix its position in 

 relation to the original axes of co-ordinates. The simplest 

 elements that can be employed for this purpose appear to be 

 the co-oi'dinates of its principal vertex [a, /S), the direction- 

 index of its axis (/*,) and its principal parameter (tt). Now, by 

 No. II. we have 



,*=— C : B=— B : A, 

 and the values of tt, a, jS will be obtained from the last three 

 equations by taking 



C cos y B B cos y A . . 



i5 cos y — C A cos y — B 



Thus, after some easy reductions, we get 



_2 (A4E— C*D) sin* y 



.(19), 



(A+C— 2B cos y)f * 



AF— Pg BD— AE / B— A cosy \ « \ 



g(BD— AE)"*" ^A2 \A+C— 2Bcosyy I 

 CF— E^ BE— CD / B— Ccosy y\"'^ '' 



2 (BE— CD) + 2C' \A+C— 2Bcosy/ J 



VIII. 



It is evident from No. ii. that the equations 



(A3/ + Ba:+D)m+(B2/+Ca:+E)=o (a), 



(A2/ + Bar+D)w'+B3/ + Ca;+E=o (b), 



denote the diameters of the curve (1) which bisect chords 

 parallel to the straight lines y =^ mf x and y ^ m x respec- 

 tively. Now the condition that the diameter (b) should be 

 parallel to the straight line y = mxis 



Kmm-\-^{m-\-m')+C — o (21) 



and this is also the condition that the diameter (a) should 

 be parallel to ^ = m'x; hence we see that if two diameters 

 (a) and (b) he so related that the first (a) bisects all chords 

 parallel to the second (b), then the second will bisect all 



