( 187 ) 



dp dp dx 



are equal to 0. From t^, — t~^, follows, if we take into considera- 

 dl dx dl 



dp dx dp 



tion tliat - - has a finite value, and — =00, that — must be 



dT dT dx 



dp 

 = 0, and also — = 0. That p lias maximum or minimum value in 



dv 



the case of an heterogeneous double point has already been repre- 

 sented by us in a drawing. (These Proc. March 25, 1905, p. 621, 

 and .lime 24, 1905, p. 184). So of the 6 differential quotients for 

 the projections of the plaitpointline 4 are zero. Two are left whose 



dp dv 



value is to be determined, viz. — and — . 



di dx 



dp 



If' we write-. — = — , we find by differentiation of numerator 



dT dT 

 dx 



and denominator 



d'p 

 dp dx'' 



df~d T f 



dx* 



dT 



dv dx 

 It' we write: — = — , we find 

 dx dl 



v) ~~drr' 



dv 



d'T 



'dv" 

 Jx, 



dv 1 



What follows may serve as a verification. Let us again write as 

 holding in the immediate neighbourhood of the double point : 



T=T ± a (x-x o y = T ± p(v-v o y 

 and 



p = p a ± y (*— *o»' — p* ± d ( y — V «V- 



For minimum value of T and p the positive sign must be chosen, 

 and reversely. So we have the following relations : 



a{x-x,Y — ii(v^v,Y, 

 and 



