818 PROFESSOR CHRYSTAL ON THE 



moves towards 0, it is obvious that OK approaches OE from one side and AB approaches, 

 the same limit from the other. 



It must be carefully noticed that the other half, OF, of the parabola (12) is the 

 limiting form for negative values of c. If this be forgotten, confusion will arise regard- 

 ing the two branches of the primitive family which pass through any point of the plane. 



Thus, for example, the two curves through P are a first branch of one primitive and 

 a second of another, both corresponding to positive values of c. At any point Q below 

 EOF the two curves are both second branches — one corresponding to a positive, the 

 other to a negative value of c : in particular, at a point Q on the dotted branch of (13) 

 the two curves are BQA and OQF. 



At the origin all the primitive curves have the same first approximation, viz., the 



parabola (12). The second approximations for any two curves c and c' are, as we have 



seen, 



rf + 12y_fcty,=<n ■ 



.^ + i2y--M c '=o S K h 



Qua intersections at the origin this pair of equations is equivalent to x 2 + 12y = and 

 X s = 0, that is to x = 0, y = thrice. Hence every integral curve is intersected by its 

 consecutive in three points at (0, 0). 



It is readily found from Newton's diagram that all the integral curves have at oo 

 the common first approximation 



<V + 3t/) 2 = (15); 



and that the second approximation to any curve c is 



(0+ 3 y y+4etf = . . . ... . (16). 



If we apply to (16) the linear transformation x — ^/rj, y= I/77, we get 



(^ + 377)2 + 4^ = 0, 



or, to the same approximation as before, 



(f+3»7) 2 -|cf=b (17). 



To the multiple point (0, 00 ) on (16) corresponds the multiple point (0, 0) of the same 

 order and species on (17). 



Now (17) has a node-cusp at (0, 0), for which 8 = 1, k=1 (see Salmon's Higher 

 Plane Curves, § 243). 



If we combine with (17) another equation 



(^ + 3 77 )2_| C '|5 ==0 (18)> 



and confine ourselves to intersections at the origin, we shall find the multiplicity of the 

 intersection of any integral curve with another consecutive or non-consecutive at (0, 00 ). 



