812 PROFESSOR CHRYSTAL ON THE 



(b o x+c o y+d o x'>)+a 1 2y = 0, . • . . (26), 



where ciid^.0, since only one value of p = when x = 0, y — 0. 



For the purpose of estimating the order of the terms we may write our equation in 

 the form 



x(b x + e y + d x' 2 ) + 01^ = ... . (27), 



the Newton's diagram for which is given by the annexed figure (3). Hence, if b d£Q, 



the approximation to the integral curve is given by b x + a 1 p = 0, which gives 

 2a 1 y + b x' i = 0, corresponding to an ordinary point. As a necessary 

 condition, we must therefore have b = ; and this is, in general, 

 sufficient; for the first approximation is now given by d^x 2 + a x p = 0, 

 which gives Sa^y + d x z = 0, corresponding to an inflexion. If we 

 translate this result into general symbols, we have as the condition for 

 an inflexion at (x, y), on the branch corresponding to the value p, 



<f> x +p<p v = 0. Hence the locus of points which are inflexions on integral curves is 



given by 



= 0, <t>*+p<f> y =0 (28), 



where p is to be treated as an arbitrary parameter, viz. , it is the tangent of the inclina- 

 tion to the x-axis of that branch of the integral curve which has an inflexion at (x, y). 



Comparing the conditions for an envelope singular solution with the result now 

 obtained we see that they may be expressed as follows : — In order that there may be 

 an envelope singular solution it is necessary and in general sufficient that the cusp-locus 

 and the inflexion-locus have a branch in common. 



Since the differential equation to the orthogonal trajectories of the family represented 



by 



<}>(®,y,p)=o, 



is 



<K x >y> -i/ot)=o, 



the inflexion-locus for the orthogonal trajectories is given by 



(P(x,y,-l/m) = , 



cj> x {x, y, - 1/w) + mfrfx, y, - 1/m) = 



or, putting p = - 1/tar, by 



0(tf> y,p) = o, p<p*( x , y, p) - <p<j(%, y,p) = o . 



Since, in general, <j> x +p<t> y =0, p<p x — < Py = Q are equivalent to ^ = 0, ^ = 0, and since 

 the p-discriminant locus is obviously the same for the original family and for the ortho- 

 gonal trajectories, it follows that at a tac-point ivhere the tangent does not touch the 

 ^-discriminant locus this locus must have a point in common with the inflexion-loci of 



