p-DISCRIMINANT OF A DIFFERENTIAL EQUATION OF THE FIEST ORDER. 811 



<t>(p>y, p)=o 



have a tac-locus, then the five equations, (p(x, p, y) = 0, (p(y, x, p) = 0, <p(y, p, x) = 0, 

 (p{p, y, x) = 0, <p(p>, x, y) = 0, have each in general a tac-locus also. 



We say, in general, because it may happen that there is not in each case a onefold 

 infinity of values of x and y, and then we have merely a tac-point. Thus, for example, 

 let us take Glaisher's Example VIII. (Math. Mess., xii. p. 6) 



0=(« 2 — a 2 )p 2 — 2xyp — x 2 = . 



The conditions for a tac-locus are (x 2 — a 2 )p 2 — 2xyp — x 2 = 0, 2xp 2 — 2yp = 0, — 2xp = 0, 

 2(x 2 - a 2 )p — 2xy = 0. These are satisfied by x = 0, y — y, p = : so that x = is a tac- 

 locus. If we interchange x and y, we get the equation 



(y 2 — a 2 )p 2 — 2xyp — y 2 =0. 



To get the tac-conditions for this new equation we have to adjoin the equations x = x, 

 y = 0. p - : y = is, therefore, a tac-locus. If, however, we interchange p and y, we 



get 



(x 2 — dF)y 2 — 2xyp — x 2 = , 



the additional tac-conditions for which are x = 0, y = 0,p=p>- Here, therefore, there is 

 merely a tac-point. 



Since a surface of the second degree cannot have a double line without degenerating, 

 it follows that an irreducible differential equation of the first order which is integral 

 in x, y, p, and whose degree in (x, y, p) collectively does not exceed the second, can have 

 no tac-locus, a result which might also have been deduced from the remark made 

 above, p. 810. 



Locus of the Points of Inflexion on the Integral Curves of the Differential Equation — 



Ao + A^-f .... +A n p n = 0. 



If, instead of considering the integral curve as a locus of points, we consider it as the 

 envelope of its tangents, we see that to a locus of cusps on the integral curves corre- 

 sponds a locus of inflexions. As the cusp-locus is present in the general case, the 

 inflexion-locus will also be present in the general ease. 



The conditions for an inflexion on one of the integral curves at any point can readily 

 be found by the methods above used. Let us take, as before, the point in question for 

 origin and the tangent to the integral in question as axis of x ; then, remembering that 

 y is of higher order than x, and omitting terms that are prima facie negligible; we may 

 write the differential equation in the form 



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



