p-DISCRIMINANT OF A DIFFERENTIAL EQUATION OF THE FIRST ORDER. 819 



Now (17) and (18) are equivalent to (£*+ 3^) 2 --*cf 5 = 0, f = 0; that is to (£ 2 + 3i7) 2 = 0, 

 g 5 = ; that is to i7 2 = 0, £ 5 = 0. Hence ten intersections of (17) and (18) are condensed 

 at (0, 0) ; and therefore any two consecutive (or non-consecutive) integral curves inter- 

 sect ten times at (0, qo ) . . . 



These results may be verified by rinding the as-eliminant of (7) and the equation 

 derived by differentiating its characteristic with respect to c, viz., 



(x o - + l2y)c-x(x* + 9y) = (19). 



The result is 



/■%..■ - C ) 3 = (20). 



Since the degree of (20) ought to be 16, we see that two consecutive integral curves 

 intersect in 3 points at (0, 0), 3 at (c, 0), and 10 at (0, oo ). 



The family of integral curves has therefore no envelope, real or imaginary ; but merely 

 the cusp-locus y = and the fixed tac-points (0, 0), (0, oo ). 



It appears, therefore, that the ^-discriminant curve in the present case, notwith- 

 standing that the primitive is algebraic, is, as usual, a cusp-locus, and is in no proper 

 sense an envelope. It might, perhaps, be contended that it is an envelope in the sense 

 that every primitive touches every other at one particular point, viz., (0, 0), or y = 0; 

 but, on the other hand, y = 0, p = does not satisfy the differential equation, and is 

 therefore not a solution at all, much less a singular solution. 



Moreover, it is clear from all that precedes that, although it is true that when the 

 differential equation 



ay -f ix 2 + cxp -\-jfi = 



has a singular solution, its primitive is algebraic; yet, on the other hand, it is the 

 exception, and not the rule, that it has a singular solution when its primitive is 

 algebraic. 



I have gone carefully into this matter, because the conclusion just arrived at shows 

 that a proposition laid down by Cayley in connection with his well known Geo- 

 metrical Theory of Singular Solutions of Differential Equations of the first order is 

 erroneous, or at least very misleading. In his second paper on the subject (Mess. Math., 

 vol. vi. p. 23, 1877) the following passage occurs: — "Consider now a system of 

 algebraic curves U = 0, where U is, as regards (x, y), a rational integral function of the 

 order m, and depends in any manner on an arbitrary parameter C, I say that there is 

 always a proper envelope, which envelope is the singular solution of the differential 

 equation obtained by the elimination of C from the equation U = and the derived 

 equation in regard to (x, y). It follows that the differential equation (L, M, N g p, l) 2 = 0, 

 which has no singular solution, does not admit of an integral of the form in question, 

 U = 0, viz., an integral representing a system of algebraic curves." 



Cayley, rests the above conclusion, which we have seen to be essentially erroneous, 

 on a demonstration which amounts practically to this : — Consider an algebraic curve 



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



