( 428 ) 



imply tluit the pülytopc iiiidei' consideration is a })risrnotope. So tlie 

 polyiiedroii t(} (tlie last poiyiicdro]i of the iirst row of li,u'. 55, I.e. ]). 189, 

 vol. II) eorresponds in symbol (24, 36, 14^ of eharaeteristic nnndiers 

 with the prism l\., ; therefore its EuLKuian function 2^-\-'Mx-\-\4:,v^--\-x^ 

 is decomposahle into the EuLERian fnnctions 12 -\- Vh; -\- x"" and 2 -\- x 

 of the l)ase and the upright edge of l\^, though ^^> is no prism at all. 



Hy |)utting x =^ — 1 we deduce from theorem I that the product 

 of the EuLERian forms of the constituents is the Erj,ERian form of 

 the prismotope. So we tind: 



Theorem. II. "A prismotope satisfies the law of Euler, if and 

 only if this is the case with all its constituents". 



If we denote as "sum of limits" of the polytope {P)n with the 

 symbol {a^, a^, . . . ,a„-\) of characieristic numbers the expression 

 (/^_|_^/^ -j- . . . -|- (/„_! -|- 1, where the last unit corresponds to the 

 polytope itself, the tii-st theorem gives for j' = l : 



Theorem 111. " The sum of limits of a i»rismotope is the product 

 of the sums of limits of its constituents". 



Corollaries. "The sum of limits of the simplotope mentioned above 

 is (2,,+i - 1) (2,,+, - 1) . . . (2'v+i - 1)." 



"The sum of limits of the yi-dimensional pai-allelotope is 3».". 



Example. The sum of limits of the jn-ismotope (/CV>; ^6V>) mentioned 

 above is 147^ = 21609. 



Physics. — "llie varlahUltii of the (/uaniiti/ h in van der Waals' 

 equation of titate, aho in connection miih the critical quantities." II. 

 By J. .1. VAN Laar. (Communicated by Pi-of. H. A. Lorentz). 



For b' we lind with x = l according to formula (11): 

 ^, ^ 7./^(l-i^)y(l +^f) ^ 0,021*32X1,227X 2,227 ^ 0^908 ^ ^^^^^^_ 



m 



1,107 1,107 



This value is — as w^as to be expected (see also I, p. 292) — 



somewhat lower than that which was formerly calcnlated with the 



eqnation of state iciihout RT being mnltiplied by the factor 1 -[- ^/?. 



1 1 



Before about was found, but now about--. 



13 ly 



For — vh" we calculate with ,/■ = 1 according to (12) : 



2,114 1,227 _ Q ()^^g.2 J-.. 454 _^_ q g^^é X (2,227)^] 



V (D 1 



V — b mr 2 



~ 1,114 1,357 



z= 1,898 X 0,9038 X 0,02162 X 7,969 

 = 1,715 X 0,1723 = 0,-J95. 



