MULTIPLE POINTS. 321 



~ 



have assumed ~ to have more than one value. But as in 

 ctoc 



this case two different branches of the curve pass through 

 the same point, there will generally be two different values 



of -T^; by Art. 181, 



dy d*<j> dy 



2 



2 ' 2 



y\* 

 xj 



dx 2 ' dx dy dx dy 2 \dx dy dx* 

 and since $(x, y) is rational, each of the differential coefficients 



of <j) has only one value ; hence if -- be different from zero 



t/ 

 72 ^2 



^ can have only one value. But, by supposition ~ has 



03? fa dx' 



more than one value ; therefore -,- = is the condition that 



dy 

 must hold at the point where two branches touch. Since 



^ = 0, it follows from (1) of Art. 298 that ^ also = 0, 

 dy dx 



d 2 ii 

 If j~ should have two equal values, then the reasoning 



d*y 

 of this Article may be applied to ~ 3 and the third derived 



equation of (f> (x, y] = ; and the same result as before may 

 be deduced. 



Points where two or more values of -f- are equal are 



uX 



called " Points of Osculation." 



300. Example. Let f - x* (1 - a; 2 ) = 0. 



Hence x = 0, y = Q, are the co-ordinates of a point which 

 may be a double point. Equation (2) of Art. 298 becomes 



'therefore ~jj- = 1, and there is a double point. 



We may in this case put the given equation in the form 



y = x */(! x*), 

 T. D. c. Y 



