102 Mr. Lubbock oti a Property of the Parabola. 



Subtracting the latter equation from the former, I find an 

 equation which may be written as follows : 



(.r, +4'2 — 2 X) {xi—Xci-\- {y^—y^ cos xy) 

 + (3/1+3/2-2 Y) (3/1-3/2 + (-^1— -^2) cosxy) = 0. 

 Substituting in this equation for 3/1 —j/.^ its value 

 j^ {x,-x^f (See p. 101). 



and dividing by x^—x^, 



(Xi +Xci—2 X) (xi yc^—x^y, + -- (x, -x^ cos x j/) 



+ (3/1+3/2-2 y)(|-(a?i-X2) + (Xij/2-X2J/,)cosj:j/) = 0, 

 which equation may be written in the form 

 {X1+X2 + X3+-I -2X 



+ ( j/i +^2 + 3/3-2 Y) cos xy} (x^yi—x^yi) 

 + ^ I {(•2^i + '^2)cosxj/ — 2 Y-2 Xcosxj/}(x, + X2) + x, j/i —X^^2 > 



(3:3 + 3/3^08x3/) (x,j/2-X2J/,) = 0. (5.) 



and also by symmetry, since tlie three points (x, ,2/,), (a;2,3/,.)» 

 (Xg , ^3) are similarly related, 



{X3 + X3 + X,+ -|- -2X 



+ (3/2+3/3 + 3/1-2 y) cosxj/} (Xgj/a — a?3^2) 

 ■*" |j{(a:2+y3) COSX3/-2 y-2Xcosx^}(x2-X3)+X23/2-«33/3| 



— (Xi+3^1 cosxy) {Xc^y^-x^y,) = (6.) 



/(jr3 + a:, + X2+ |- -2X 



+ (3/3 + 3/1+3/2-2 Y) cos xj/| (X3yi -X, J/3) 

 + Y 1 {('r3 + a:,)cosxj/-2Y-2Xcosxi/}(x3-x, +{x^y^—x^y^ l 



-(•^2+3/2Cosxj/) (X33/1— Xij/a) = 0. 

 Adding together the three last equations, many terms de- 

 stroy each other, and 



