p-DISCRIMINANT OF A DIFFERENTIAL EQUATION OF THE FIRST ORDER. 817 



For the first branch y' vanishes when x = and when x = f c ; these give a maximum 

 turning-point at the origin and a minimum turning-point at (f c, — ^c 2 ). 

 At x = c there is obviously a cusp, the value of y' being ^c. 

 Near the origin, approximation to the first branch is given by 



y=-^ 2 +2 1 



x° 



(ii); 



from which we see that all the first branches osculate the parabola 1 2y + x 2 — at the 

 origin ; and depart less and less from it the more we increase c. 

 It is also obvious from (7) that this parabola 



12?/+.^ = o 



(12) 



is the limiting form of the integral curve corresponding to c = oo ; and also that it 

 divides the region below the x-axis into two districts, in the upper of which the two 

 values of c corresponding to the two curves of the family which pass through a given 

 point are of the same sign in the lower of which of opposite signs. 



The following figure, therefore, gives a sufficient representation of the forms of the 

 curves of the family for positive values of c. For negative values we have simply to 

 reflect the diagram in the y-axis. KOAB, figure (4), is the curve corresponding to any 



Fig. 4. 



finite value of c, K'OA'B' that corresponding to a greater value. The parabola GOH is 

 the limiting form corresponding to c = oo , the second branch of which is altogether 

 at oo . The limiting form corresponding to c = + is the left-hand part of the parabola 

 EOF, whose equation, as is easily seen from (7), is 



3y+« 2 = (13). 



We have thickened this part to indicate that it must be reckoned twice ; for, as A 



