p-DISCRIMINANT OF A DIFFERENTIAL EQUATION OF THE FIRST ORDER. 823 



c s (x + y)z 5 — Sc 2 xyz 2 + xhf = 0, "1 /-.<>-, 



c(x+y)z — 2xy = ) 



This last is equivalent to 



c\x + y)z % - Z&xyz 2 + \c 2 (x + yfz 2 = 0, 



= °'j .... (14). 

 c{x-\-y)z — 2xy =0 ) 



(14) is equivalent to z 2 =0, xy = Q, which gives (1, 0, 0) and 0, 1, 0) each twice, together 

 with 



c(x+y)z-2xy+l(x+yy = Q,\ , 



c(x+y)z — 2xy—0 ) 

 that is, 



that is, 



(*-y?-*>\ ...... (16); 



x+y)z—2xy=0 J 



(x-yy=o; 



n'l • (17) - 



x(cz — x) = ) 



Now (17) gives (0, 0, 1) and (c, c, l) each twice. Hence the intersections are (1, 0,0) 

 six times, (0, 1, 0) six times, (0, 0, 1) twice, and (c, c, 1) twice. There is, therefore, a 

 node locus, but no envelope. 



As a farther verification we may calculate the c-discriminant of (11). The result is 



27z*xiyt(x-yy (18). 



The factor (x—y) 2 corresponds to the node locus, the factors x i , y i , z° to the sides of the 

 triangle of reference, which are parts of degenerate transition quartics of the system. 



By means of a conic passing through A, B, C and the acnode, it is easy to obtain 

 rational expressions for the ratios of the coordinates in terms of a parameter 6. The 

 result is 



xjy=-2c8(0 + c)/(e-cy, } 



ylz=-2c6(6-c)l(6 + cy | ' • * ' W> 



by means of which this interesting family of curves may readily be traced. 



It would be easy to construct a large number of examples of the degeneration of the 

 envelope of an algebraic family. It will be sufficient to add a pair of examples of cubics 

 which have this property. 



The cubic 



z 3 + tfxy(x + y) — 3c 2 xyz = 



has real inflexions at (0, 1, 0) and (1, 0, 0), the inflexional tangents being x = 0, y = 

 respectively, passes through the fixed point (1, —1, 0) and has an acnode at (l, 1, c). 

 Any two consecutive curves of the family intersect thriee at (0, 1, 0), thrice at 



VOL. XXXVIII. PART IV. (NO. 24). 5 Z 



