36 Proceedings of the Royal Irish Academy. 



tangentsj to a linear complex, the tangent line at such a point has three-poiat 

 contact with the curve, as in the case of an inflexion on a plane curve. Such 

 a point we shall term an infA^d^yti in the course of this paper. 



5. M. Picard's theorem may be extended to higher singularities as 

 follows : — 



If at o^ point P on a curve ielonging to a linear coraple-x the curve has p, §, r 

 consecutive points in common wiih the osculating plane, the tangent, and an 

 arbitrary plane through P respectively, then shall 



p = g^ r. (3) 



To prove this, we take the osculatiog plane at P as rt'i = 0, the plane at 

 infinity as rcj = 0, the plane through the tangent at P, and the pole Q of the 

 plane at infinity with respect to the complex as a;^ = 0, and any other plane 

 through PQ. as iCg = 0. 



The equation of the complex will be of the form 

 aui^u + a-i^fi-i = . . . 



For a tangent to the curve we can write 



hence the coordinates {x) of a point on the curve satisfy the equation 



— Orn dxi -T aa(x2 dxs — »^ dx^) = 0, 

 when we have put x^ = 1, (fo;4 = 0. 



Xow in the neighbourhood of P, in virtue of what is given, we can express 

 the coordinates of a point on the curve in series of ascending powers of a 

 parameter t in the form 



Xi = atP + . . . 

 Z2 = bt^ + . . . 

 X3 = ct" -~ . . . 



2:4 = 1 (abe ^ 0, p > q > r) . 



Substituting these values in the differential equation satisfied by the curve, 

 the terms of lowest degree on the left-hand side are 

 - a„ aptP-' -a^bc{q-r) P'^-K 

 We must therefore have, if the equation is to be satisfied identically, 



p = q -T r. 

 6. This form of the proof is probably the simplest that can be given ; it is 

 substantially the same as that given by M. Picard for the restricted form of 

 the theorem (p = 4 j = 3, r = 1). 



The argument may, however, be put in another form, which has the advan- 

 tage of showing that the theorem is only a particular case of the reciprocal 

 property mentioned above (ii). It also introduces considerations to which we 

 shall have occasion to recur. 



(4) 



