AXM.YIK- 9DOMKTHY [Cii. VIII. 



\\liidi is tin- equation of the secant line LM of the given 



ic (1). 



If no\\ this secant line be revolved about J\ until /' 

 mmes into mineidener with /' p /.., until Xi=r\ and //-.. = //!, 

 this equation beeom 



whieh is. therefore, the equation of the tangent line /', 7 f at 

 the point P,. This equation (8) can be put in a mneh 

 simpler and more easily remembered form, thus : 



Clearing equation (8) of fractions, and simplifying, it may 

 be written 



... (9) 



lut, from equation < I . 



Ax? + By? + Ox l + F^ = - Gx l - Fy v - (7, 



hence substituting this value in the second meml>er of equa- 



tion (!) that equation becomes 



Ax\x -f By } y + G-x + Fy = O-x l Fy l (7, . . . (10) 



and, by transposing ami combining, this may be written, 



This is, then, the equation of the tangent to the coni< 

 Ax 2 -f By* 4- 2 Ox + 2 Fy -f C = 0, 



whatever the values of the coefficients A^ B, 6?, F, and 

 may be ; the point (* y^) beinu r the point of contact. 



i=0, B=l, G= . f'=n and (7=0, then the equa- 



tion of this conic becomes y 2 = 4 px, and the equation of the 

 tangent becomes, y l y=2p(x + x\)> similarly for any ol 



.1 form of the equation of the cnnie. 



* Compare note, Art. 84, (/?). 



