84 Mr. Lubbock on a Property of the Conic Sections. 



Equating these values of a, we obtain the following equation 

 of condition, upon which the truth of the theorem in question 

 depends, 



C^i— 5^4) { (^s-^e) (^5i/2— 3/5 ^2) + (^5—^2) {^6^3-2/g ^3) } 



+ (^2— 5^5) { (-^i —•^4) (^65^3-^6 -^3) + (-^6— -^3) (^4^1 -3/4 ^1) } 



= 0. 



It remains to prove 'that this equation does hold good, in 

 consequence of the relations which exist between the quan- 

 tities contained in it. 



We may make Xi = 0, .Tg = without limiting the gene- 

 rality of the theorem, the direction of the coordinate axis y 

 being any whatever ; this amounts to taking for the origin the 

 point in which the line 1 2 touches the parabola. In this 

 case the points 5 and 4, 6 and 3, 1 and 2 are symmetrical 

 and similarly involved, and the preceding equation of con- 

 dition becomes 



(3/1 — .2/4) -^5 {-^3 (3/2-3/6-) + -^6(3/3-3/2)} 

 + ( 3/2—3/5) ^4 {^6 {2/1 -3/3) + -^3 (3/6-3/1) } A. 



+ (3/3-3/6) •^4 ^5 (3/2-3/1) = 0- 



The equation to the tangent of the parabola j/^ — p x pass- 

 ing through the points (2, ?/) («, jS) is 



p[x—oif-^y[x-'a){y-^)-\-^x{y-^f = B. 



If x,y coincide with the point 2, so that Xc^ = 0, and if 

 a, /3 coincide with the point 3, 



SMarly x« = iMfcg.) . 



Again, if in the equation B the point (^,j/) coincide with 

 the point 3, the two values of —— ^ correspond to the direc- 

 tions of the lines 3 2 and 3 4. 



^ _ 23/3 + ^"Z yi~px^ 



3/3-/3 jp 



Making a = a^g = 0, i3 = ^2 > ^^^ taking the upper sign 



^ - (3/3-3/2) (2j/3-2 ^ yi-'^y^{y^-y'^ 



= ^.^2(3/3-^2) ^g {jeforg^ 

 V 



