ORDINARY LINEAR DIFFERENTIAL EQUATIONS 



273 



integration. The differential equation belonging to (11) has the form 



/A,+j9B,+C.=0, (12) 



and the ^-discriminant is 



Bi^-4A,C,=0. (13) 



Determining A,, B,, C, from (11) in the usual manner and substi- 

 tuting them in (13) the result is 



(B='-4AC) 



(14) 



If B^ — 4AC=0, then for every point of this locus the two X'*, i. e., 

 the two integral curves through this point coincide. In other words 

 B' — 4AC==0 is the envelope locus of (11). 



For a point of the locus represented by the determinant, the two 

 \'s in (11) are different, i. e., at such a point two distinct integral 

 curves are tangent to each other. The locus represented by the de- 

 terminant in (14) is therefore the tac-locus, and it clearly appears 

 twice in the j9-discriminant (14). 



4. This critical and historic account on the j9-discriminant 

 would not be complete without a few remarks concerning the seem- 

 ing discrepancy between Darboux's and Cayley's results. 



Darboux does not assume any knowledge on the system of in- 

 tegral curves of a given differential equation, i. e., he considers the 

 most general equations without reference to their possible origin. 

 Cayley however considers only such equations, which traditionally 

 are formed from a given system of curves, / (.t;, y, c) = 0. In other 

 words he assumes that every differential equation admits of such a 

 specific system, having an envelope generally. 



The fact is, as Picard states, loc. cit., that to an arbitrary differ- 

 ential equation corresponds a family of curves which from the point 



