320 MULTIPLE POINTS. 



dii 

 equation (1) it follows that -, - cannot have more than one 



value unless both 



d$ , d$ 



-f = 0. and ~ = 0. 

 ax ay 



These then are the conditions for the existence of a mul- 

 tiple point. If values of x and y can be found which satisfy 

 these equations and the equation to the curve,, then for such 

 values of x and y we have, by Art. 181, 



dty d*<f> dy d>AfyY_ 



dx 3 dx dy dx dy* \dx) ~ 



From this quadratic equation we can find two values of -~ , 



and thus determine two tangents which can be drawn through 

 the multiple point. In this case the multiple point is called 

 a double point. 



If the above equation assumes an indeterminate form by 



the vanishing of ~ , -,?-, and -j-%, for the values of 

 dx* dxdy dy* 



x and y under consideration, we have, by Art. 184, 



.. ..(3). 



dx 3 dx* dy dx dx dy* \dx dy* \dx 



This cubic equation gives three values of -j- ; if they are 



all real, three, tangents to the curve pass through the point 

 under consideration ; the point is then called a triple point. 

 If the equation (3) assumes an indeterminate form by the 



du 

 vanishing of the coefficients of the different powers of -j-- , we 



must proceed to the fourth derived equation from <f> (x, y] = 0, 

 and we thus obtain a biquadratic equation for determining -^ . 



299. If the two values of -^ furnished by equation (2) of 



Art. 298 are equal, the two branches which pass through the 

 point in question have a common tangent at that point. 

 In this case, however, the method by which we have arrived 

 at equation (2) is not satisfactory, because in obtaining it we 



