( 366 ) 



The (Mirvo exlending iipwai-ds IVoiii P sliotild llicrcforc be bciil in 

 siicii a w;\y, Uiat its initial dii-ection was the same as that of the 

 ciiivo of the three-phase-pressure. 



The tangent plane to the (y^ T, .ly) surface being normal to the 

 plane of the %ure, because it contains a line which in P is 

 normal to the figure, every curve on that surface, passing through 

 P, will have its projection in the section of the tangent ])lane Avith 

 the plane of the figure; and so both the curves extending upwards 

 from P and those extending downwards, will \\ix\Q their projections 

 in this same sectioii. This follows also from the values of p. 241 

 (preceding connnunicationj. We have for tlie three-jdiase-pressure: 





hi the immediate neighbourhood of a plaitpoint — -— and 



X. 



is e(pial to zero, (Cont. II. )». 125;; and we find -77, = ( ^r~ I- 



(II \oiy, 

 One more remark to conclude with. Xow that we have concluded 

 to the existence of the tops of the curves Vs/ =:z and ]r,f = 0, 

 we shall also have to accept the conclusion, that the complications 

 in the course of the (/>»,. r) and the (p, T) sections of the surface of 

 fluid phases coexisting with solid ones, remain restricted to the 

 neighboiii-hood of the critical phases. It is therefore uncertaiji, whether 

 iu a section foi- gi\en ,/•, if the latter is e.g. chosen halfway betwceji 

 .t'c and .I'a, the \\\o vertical tangents still occur. As soon as they 

 have coincided, the section has no longei' any sjiecial point, and so 

 the retrograde soliditication has also disai>peare(l. 



Mathematics. — ''Ccnfrlc (Iccoiniiosiflon oj' itohjiaiH's.'" \\\ Pi-of. 



P. H. SCHOUTK. 



In the following lines it will be shown how a i-egulai" ])olyt()pe 

 can be decomposed according to its vertices or to its limiting spaces 

 of the greatest numl)er of dimensions into a system of congruent 



regulai- polyto|)es with a commoJi centre. For this /-*„,„ shall re- 

 present a i-egular ]K)lytoj)e, limited l)v m spaces /S',, _, in /S',,, with a 

 length r of the edges; and moreover we shall omit as nuic'i as 

 l)0ssible the number n of the dimensions and always each of the 

 predicates "regular", "congruent" and "concentric". 



