326 SINGULAR POINTS. 



305. A point saillant is a point at which two branches of 

 a curve meet and stop without having a common tangent 



/v 



Example. Let y = j 



e*. 



,. dy 



therefore j <z = 



+ e* 



Here, if x be positive and approach zero as its limit, we have 

 ultimately y = and ~- = ; but y 



if x be negative, we have ultimately 



dii 

 = Q and - = 1. Hence at the - 



origin two branches meet, one 

 having the axis of x as its tangent, 

 and the other inclined to the axis 

 of x at an angle of 45. 



Branches Pointillees. 



306. If a curve has an infinite number of conjugate points, 

 that series of points is called a tranche pointilMe. 



For example, suppose y = #sin 2 c; for all positive values 

 of x there are two possible values of y, but when x is nega- 

 tive y is impossible, unless # be a multiple of TT. Hence we 

 have an infinite number of conjugate points lying on the axis 

 of x and forming a branche pointillee. 



EXAMPLES. 



1. If a s y = aV x* there is a multiple point at the origin. 



2. In the following curves there is a point of inflexion at 



the origin : 



y = sin x ; y = x cos x ; y = tan x ; y x* tan x. 



3. The following curves have cusps at the origin : 



