1152 



along the curve from point to point and either in the same or in 

 opposite direction. 



When we determine, therefore, e.g. the temperature of inversion 

 and pressure of inversion of rhombic in monoclinic sulfur, or of 

 two modifications of KXO^ etc. in a mixture of two solvents and 

 under its own vapourpressure, this T and P of inversion change 

 with the com})Osition of tiie solvent. These changes are, however, 

 very small, as it follows from tiie ju-evious considerations. 



{To be continued.) 



Mathematics. — "On the singular solutions of ordinary and 

 partial differential equations of the f rst order'. By Prof. Hk. uk 

 Vries and G. Schaake. 



(Communicated in the meeting of March 28, 1914). 



Introduction. If the complete integral of a partial dilFerentiai 

 equation of tlie first order with two independent variables, F {.x,y,z,p,(j) 

 =: is represented by f{.v,j/,z,Cy,c^) = i), and if the i-esult of the 

 elimination of c, and c, from the three ec^uations 



is called for the sake of brevity E = 0, the following pectiliar 

 phenomena may arise. If the general solution ƒ (,i',7/,c) = 0, of an 

 ordinary differential equation of the first order F {.i',y,i)) = 0, possesses 

 a nodal locus, it belongs generally speaking to the result of elimination 

 of c from the two equations 



oc 

 and only in o?ie particular case it does not belong to it ; with the 

 partial equations it is just the reverse, at least if in this case the 

 locus of the nodes consists of one or more curves; if there is a 

 nodal surface, it does belong in general to F=0, though there is 

 a possibility that it does not. 



It is a matter of course that all possible cases may be arrived at 

 by a purely analytical method ; but it appears that considerations 

 derived from polydimensional geometry throw a vivid light on those 

 various analytical possibilities and so to say increase the differences 

 and render them more essential ; to prove this is the aim of the 

 following paragraphs. 



§ 1. Let in the first place be given an ordinary differential equation 

 of the 1**' order 



