820 PROFESSOR CHRYSTAL ON THE 



U = of order m, having singularities equivalent to 8 double points and k cusps. Of 

 the intersections of U = 0, with its consecutive U + S C U = 0, two will coincide with each 

 double point and three with each cusp, leaving m 2 — 28 — 3k other points of intersec- 

 tion. If n be the class of the curve m 2 — 28 — 3K = m + n, a number which cannot 

 vanish ; hence there is always an envelope, viz., the locus of these, m 2 — 28 — 3/c points. 



The fallacy in this reasoning consists in assuming that the m 2 — 28 — 3k points are 

 necessarily spread out into a locus. It is, in fact, an inference that may be drawn from 

 the above investigation that, in general, when a differential equation of the first order 

 has an algebraic integral, this is not the case. So that only particular kinds of algebraic 

 families can be integrals of equations of this description. 



If we examine the particular case above discussed, for which m= 4, 8 = 1, k = 2* we 

 see that m 2 — 28 — 3k = 8. The eight points which ought, according to Cayley's theory, 

 to form the envelope are concentrated, three at (0, 0) and the remaining five at (0, oo ). 



It is somewhat surprising that Glaisher (Mess. Math., xii. p. 2, 1883) seems to 

 endorse the statement of Cayley just referred to, seeing that Glaisher's examples (iii), 

 (xiii), (xiv), (xv), (xx)t are instances to the contrary. It might be inferred from the 

 somewhat guarded language used by Forsyth (Differential Equations, § 30), and 

 from his reference to Cayley, that he attaches some value to Cayley's result ; but he 

 has been kind enough to inform me that he is not to be understood as endorsing the 

 proposition in dispute. 



Trinodal Quartic Family ivhich has no Envelope. — Inasmuch as the quartic family 

 already discussed presents a peculiarity in respect that there is a condensed singularity ; 

 it may be well in the interest of the theory of envelopes to show that the degeneration 

 of the envelope is not due to this cause. This we shall do by constructing a trinodal 

 quartic whose singularities are all distinct and which has no proper envelope. The 

 possibility of this may be seen d priori by considering that we can construct a quartic 

 which has cusps at two given points, B and C, and given cuspidal tangents at these 

 points, which passes through a given point A, and has a given tangent at A : this 

 involves 5 + 5 + 2 = 12 conditions. That there be a third node (position not specified) 

 involves one more condition, leaving still one degree of freedom ; so that we have a 

 family of quartics fulfilling the given conditions. Now (as may be seen by discussing 

 the intersection at the origin of the curves y 2 = ex 3 , y 2 = c l x 3 J) any two (and therefore 



* The curves are unicursal quartics : by considering the intersection with the parabola Zy = /*x(x - c) we find 



x = 4f*cl(l+tcy 

 3y=-V(i_^)V(l+^. 

 + There is an oversight in the interpretation of No. (xx). 

 \ The two equations are equivalent to 



y 2 = ex 3 , 

 (c_c 1 ).c 3 = 0; 

 that is to 



■ ; = 0, 



tf=Q, 



which gave x-0, y=0 six times. 



