Mr. Lubbock on a Property of the Conic Sections, 85 



Taking the lower sign and making a, /3 coincide with the 

 point 4. 



"'4 — 1^ * 



4 p 



Similarly, because the points (5, 4), (6, 3) and (1, 2) are sym- 

 metrical, 



^ _ ^{yG-yi){yo-y6+yi) 



b p 



Again, by equation B, 



x-a; = (-^-3/4) {^.^4 ± 2 a/ 3/4^-/? ^4} 



if the upper sign refer to the direction of the line 4 5, the 

 lower refers to the direction of the line 4 3. 



r - ^ (i^4-i/3 + 3/2) (.^5-3/4 + .^3-3/2) 



Equating this value of x^ to that found above, 



(3/4-3/3+3/2) (3/4-3/3 +3/2+^5) +(^6-3/1) (3/5-3/6+ 3/1) - 



,j 0, 4-w - 3/5 ± ^3/5^-^ (3/6-^1) (3/5-3/6+3/1) 



y\~yz'^y'i— 2 • 



Hence y^—y^ + 3/2 is equal to 3/5-3/6 + yi, or to y^-y, . 

 Employing the values of o^g, ;i"4, Xc^ and j;g which have been 

 found, and the remarkable equation of condition 



3/4-3/3 +y2=y6 = ^6+ ^1 or 

 3'i-3/2 + 3/3-3/4 + 3/5—3/6 = 

 it is easy to verify the truth of the equation A, p. 84. 

 This equation may now be put in the form 



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

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

 + (3/3-3/6-) (3/3-3/2) (3/4-3/3+3/2)' (3/6-3/1) (^2-3/1) 



= (3/6-i/O (3/3-^2) (3/4-3/3+3/2) {(5/1-^4) {3/2(3/2-3/6) +3/1 (3/6-3/1)} 



+ (3/2-3/5)13/1 (3/1 -3/3) +3/2 (3/3-3/2)} +(3/3-3/6) (3/4-^3+3/2) 



(3/2-3/1)} 



putting for ^2—^/5 its valueyg— 2/4+^,— ^g, the quantity be- 

 tween the brackets becomes 



(3/1-3/4) {3/2 (^2-^6) +yi (y6-3/r)i-^ 



+(^3-3/4+3/1-3/6) {yi (yi-y3)+3/2 (y3-y2)} 



+ ( 3^3-3/6) ( ^4— y3 +- ^2) (3/2—3/1) ^-^ 



= (yi-3/4) {3^2 (3/2-3/6) +yi (^6-3/1) +yi (yi-3/3) +3/2 (3/3-5^2)} 



