808 PROFESSOR CHRYSTAL ON THE 



It remains to consider the nature of the p-discriminant near the origin. For this 

 purpose it is necessary to retain more terms in the differential equation, for y and x are 

 now of the same order of magnitude. Since p is small, the following equations will be 

 sufficient for our purpose : — 



d x 2 + e Q xy +f if + (\x + c t y)p + a 2 p* = 0, 

 b 1 x+c 1 y+2a z jp- 



(19). 



! =M- • • • 



. = 0. j 

 Eliminating p, we have for a first approximation to the _p-discriminant 



^ald,x^e (j xy + Uf)-{\x + c 1 yf = Q . . (20). 



This indicates that the ^-discriminant has a double point at the origin. Hence, the 

 conditions that two integral curves touch each other and do not touch the ^-discrimi- 

 nant at any particular point are b = 0, c = 0; and at such a point the ^-discriminant 

 has a double point. 



In order that the ^-discriminant, or a branch of it, may be a tac-locus, the conditions 

 b Q = 0, c = 0, must be satisfied at every point in question. Since it is impossible that 

 every point of a continuous irreducible curve can be double, it follows that the 

 j)-discriminant must be reducible, must in fact contain a squared factor whose square 

 root is the characteristic function of the tac-locus. 



Hence, when the ^-discriminant furnishes a tac-locus, the two conditions b = and 

 c = must be satisfied at every point of it. Its characteristic then contains a squared 

 factor whose square root is the characteristic of the tac-locus. 



General expressions for the conditions for the existence of an envelope singular 

 solution or a tac-locus can readily be obtained by transforming the differential equation 



0(X,Y,P)eeA o +A 1 P+ + A n P» = 0, 



where the co-ordinates are (X, Y), and P = dY/dX, to the tangent and normal at the 

 point (x, y) as axes. If the new co-ordinates be (£, n), ■& = drijd£ > , and X= l/*/(l +P*)> 

 f x= PlJ{^ +.P 2 )> we nave X = x + A£ - m, Y = 2/ + /"£ + H and J > = (p + m)/(l -pw) = 

 p + (l+p 2 )m+p(l+p) 2 W+ . .. =ff + «CT + foar 2 + .. . , say. 

 The differential equation then becomes 



•#B + A£-/«fc y+v£+~K>i,2>+am + /3w*+ ) = . . (21). 



This gives to a sufficient approximation for our present purpose 



+ 20„(X|- M)(v-i+ \v) + 2&«(X£- MXaJS + fa' 2 ) 



+ 2^Xu£+X7 ? )(aW + /3*J 2 )] = . . (22). 



