CONJUGATE POINTS. 

 may be possible at a conjugate point ; for example, suppose 

 3,= *Jv(*'-a'). 



Here, when x 0, we have ~ ; but the origin is a con- 



jugate point, since c = 0, # = 0, satisfy the equation, and y 

 is impossible for all other values of x between a and a. In 



d\ 

 like manner -r4- or any number of the differential coefficients 



of y may be possible at a conjugate point, but they cannot be 

 all possible, for if they Avere we should have nothing to dis- 

 tinguish the point in question from an ordinary point of the 

 curve. 



To find the condition for the existence of a conjugate point 

 Since at a conjugate point the values of the differential 

 coefficients of y cannot be all possible, let the n 01 differential 

 coefficient of y be the first that is impossible. Suppose the 

 equation to the curve to be put in a rational form, and 

 denoted by < (x, y} = 0. Take the n t:i derived equation ; we 

 have 



da? 



where the terms not written down contain differential coeffi- 

 cients of < with respect to x and y, and also differential 

 coefficients of y with respect to x of orders inferior to the n th . 



If then -r- be not zero the value of -=-* furnished by the 

 dy dx n 



above equation will be possible ; hence -^ = is a necessary 



y 

 condition for the existence of a conjugate point ; but 



dx dy dx 



therefore also p- = 0. 



dx 



303. It appears from the preceding Articles that if 

 < (x, y}=Q be the equation to a curve, we must have at 



