246 



As to the number of isomerides which can be expected when the 

 number of asymmetric and pseudo-asymmetric carbon-atoms in 

 the molecule is known, the following data may suffice. 



If n be the number of true asymmetric carbon-atoms in the mole- 

 cule, N a the number of the possible optically active isomerides, 

 Ni that of the possible inactive and non-resolvable isomerides, and 

 N r the number of racemic compounds, we have in the various cases 

 the following relations: 



a) If no reduction of the number of isomerides occurs, neither 

 by "internal compensation", nor by the presence of a pseudo-asym- 

 metric carbon-atom in the molecule, then generally: 



N a = 2 n , and N r = \ N a = \. 2 n , while Ni = here. 



b) If internal compensation occurs, without the influence of a 

 pseudo-asymmetric carbon-atom, then: 



N a = 2 n - J , N r = $. 2"- 1 , and N t = 2%~ 2 - 



c) If there be a pseudo-asymmetric carbon-atom, these numbers 

 become : 



N a = 2?- 1 2~*~', N r = %(2 n - 1 2~2~);a.ndN i = 2~*~. 



22. In the case considered here, the impossibility of the fission 

 of the meso-tartaric acid and of the two inactive trioxyglutaric acids 

 was an immediate consequence of the existence of a symmetry- 

 plane in their atomistic arrangement. The same however must 

 occur if the arrangement have a mirror-axis or a symmetry-centre 

 among its symmetry-elements. Such cases can occur, as soon as 

 asymmetric carbon-atoms are units of a cycle of atoms. A few selected 

 instances may further explain this. 



If there be only a single asymmetric carbon-atom in the ring, 

 the influence produced by that atom is in principal features the 

 same as that in open-chain compounds with a single asymmetric atom. 

 In such cases the plane of the ring can of course never be a symmetry- 

 plane of the molecule, and the number of isomerides is two, not 

 counting the racemic compound. If however there be two or more 

 asymmetric carbon-atoms, more detailed examination is necessary. 



Ladenburg 1 ) drew attention to the fact that the trans-modiii- 

 cation of the diketo-piperazines (A), is not resolvable into enantiomor- 

 phous isomerides. He, and later Groth *), explained this by indi- 



i) A. Ladenburg, Ber. d. d. Chem. Ges. 28. 1955, 3104. (1895); P. Groth, 

 ibid. 28. 2510. (1895). Cf. the note on page 244. 



