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dT . 



double poinl — is nol equal l<> 0, is also immediately seen when we 



dp 



dT 

 consider I hat tor such a point also an = 0, and — appeal's there- 

 dp 



fore in an indefinite form, the value of which we shall presently 



dv f dv \ 



determine. Hence — = for such a double point, which is not 



dx \ dx Jp 



the case for an homogeneous double point. 



Of the 6 differential quotients which come in for discussion, three 



are equal to for an homogeneous double point, and three are left 



dv dv dp 



the value of which is still to be determined, viz. — , — and — . 



dx dp dx 



If we write 



dT 

 dv dx 

 dx~dl' 



dv 



dT 

 dv dp 

 dp dT 



dv 



and 



d'J 

 dp dx 

 dx^df' 



dp 



in all these three expressions both numerator and denominator is 

 equal to 0. If in the first we differentiate numerator and denomi- 

 nator with respect to x, in the second with respect to v, and in the 

 third with respect to p, we find : 



d s T 

 dv dx* 



dx d* I 1 dv 

 dv* dx 

 d*T dp 

 dv dp' dv 

 d~p~~ d l T 

 ~dv* 

 and 



