TANGENT AND NORMAL. 283 



The length of the perpendicular on the tangent from the 

 origin is, neglecting the sign, 



x 



263. It is sometimes convenient to determine a curve by 

 the two equations 



so that oc and y are both functions of a variable t, by elimi- 

 nating which between the given equations, a result of the 

 usual form y f (x) may be obtained! With this supposition, 

 we have 



dy 



dy _ dt 



dx dx' 



Hence the equation to the tangent becomes 



('-} -fo'-*^ 



* y} dt (x X] dt ' 



and the equation to the normal becomes 

 / / \ dy . , .dx 



<-)--(-) 5 . 



In the figure we have supposed the axes rectangular; 

 if they are oblique no change is made either in the inves- 

 tigation of the equation to the tangent or in the result. But 

 the equation to the normal is 



dy 

 1 + cos o> -, - 



t dx i . 



y -y= -- %~ (* -*). 



COS (0 + ~- 



dx 

 where ta is the angle of inclination of the axes. 



264. Example (1). The general equation to a curve of 

 the second order is 



Ay* + 2Bxy +Cx*+ 2Dy + 2Ex + F= 0. 



