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XXIV. — On the ^-discriminant of a Differential Equation of the First Order, and on 

 Certain Points in the General Theory of Envelopes connected, therewith. By 

 Prof. Chrystal. 



(Read 15th June 1896.) 



The theory of the singular solutions of differential equations of the first order, even 

 in the interesting and suggestive form due to Professor Cayley (Mess. Math., ii., 

 1872), as given in English text-books, is defective, inasmuch as it gives no indication 

 as to what are normal and what are abnormal phenomena. Moreover, Cayley added 

 an appendix to his theory regarding the circumstances under which a singular solution 

 exists, which is misleading so far as the theory of differential equations is concerned, if 

 not altogether erroneous. 



The main purpose of the following notes is to throw light on the point last men- 

 tioned by means of a number of examples. I have also taken the opportunity to 

 furnish simple demonstrations of several well-known theorems regarding the p-discrimi- 

 nant which do not find a place in the current English text-books. 



It may be premised that in what follows we shall regard only such integral curves of 

 the differential equation at any point as admit of an approximate representation of the 



form 



y = \x a + /xxP+ 



where a, /3, .... are commensurable numbers, i.e., only integrals which have an 

 ' algebraic point ' at x, y. 



Nature of the Integrcd of the Equation — 



A + A 1 2) + A,p*+ +A„y ! = .... (1) 



at a Point on the ^-discriminant Locus, in the most General Case. 



We suppose that at the point in question A , A l5 , A n are synectic : so that 



we have 



A = a + b x + c y + d & + .... 



A 1 =a 1 + b 1 x+ c 1 y + d 1 x 2 + 



A. 2 =a. 2 + b 2 x+ c . 2 y + d 2 x*+ , } (2). 



A„ = a„ + b H x + e n y + d„x 2 + 



We take the most general case, and suppose that two values of p, and no more, 

 become equal at the point on the ^-discriminant locus. Let this point be taken as 



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



