270 UNIVERSITY OF COLORADO STUDIES 



In what follows next I shall give a direct analytic proof for the 

 dualistic relation between the resultants of the equations 



<i>{^,y,p)=0, (3) 



4>{^,y^p)=^, (5) 



8<f) 



and of 



2.<') If in 



8p 



=0. (6) 



X and y are replaced by the co-ordinates of a point in the (a;, y) 

 plane and (1) is solved with respect tojt?, then these values oi p de- 

 line the tangents of the integral curves through this point. But 

 instead of considering the integral curves through a given point we 

 may ask for the integral curves which touch a given straight line. 

 In other words, every differential equation may be interpreted geo- 

 metrically in a twofold manner, according to the principle of duality. 

 Now, the locus of the points on those straight lines at which 

 two points of tangency of integral curves with these lines coincide^ 

 is given hy the resultant of 



</> (^s y^ P)=^^ 



and 



d,r by 



and rej/resents the locus of the points of inflexion of tht integral 

 curves. (See Darboux, loc. cit.). 



To get the dualistic interpretation of this theorem we make a 

 transformation by reciprocal radi of the differential equation and its 

 sytem of integal curves with respect to the circle 



(') This section has been presented to the American Mathematical Society, at the annual 

 meeting: in Boston, Anerust, 1903. See Bulletin of the American Mathematical Soci- 

 ety, 2nd Series. Vol. X. pp. 137-139. Dec. 1903. 



