105 



character, and experience has readily confirmed this conclusion. 



We may also ask: to which and to how many classes will the 

 phenomena of pyro- and piezo-electricity appear to be restricted, 

 phenomena for which the absence of a symmetry-centre appears 

 to be precisely the striking feature? 



According to a theory of W. Voigt 1 ) on pyro- and piezo-electric 

 phenomena in crystals, in which theory the electric momentum in 

 such crystals is thought to be determined by the deformations which 

 are the consequences of the temperature-changes or of the compres- 

 sions or dilatations to which the crystal is subjected, the said pheno- 

 mena may occur in twenty of the 32 crystal-classes : of course they 

 will not be manifested in the eleven centrically-symmetrical crystal- 

 types just mentioned above, or in the crystals of the group K, which 

 do not possess any heteropolar axes. In the remaining groups 

 such dielectric polarisation may occasionally occur, if circumstances 

 are advantageous; and the difference of potential can then manifest 

 itself at both ends of any heteropolar axis. 



In an analogous way we can answer the question: to how many 

 symmetry-classes will the number thirty-two be reduced, if the physi- 

 cal phenomena considered should be described by means of an 

 "image" /, having the shape of an ellipsoid? Such is the case in the 

 phenomena concerning the propagation of light-waves, of heat, of 

 electric currents, of magnetic induction, etc. 2 ). The number of the 

 possible symmetry-groups will then appear to be reduced still more, 

 as is universally known to every mineralogist with respect to the 

 optical properties of crystals. 



15. Something analogous to what was said in the case of phy- 

 sical phenomena occurring in crystalline media of a certain symmetry, 

 will be the case if two physical causes, each having its own symmetry- 

 character, be superposed in such a way that each of them can con- 

 tribute its share to the resulting effect. The complete cause will then 

 act as having only the symmetry-elements which are common to 

 both component causes. The symmetry of the resulting effect will thus 

 also be generally of a lower degree than that of each of the causes 

 separately; but as we have already mentioned, this need not always 

 be the case, the effect having possibly also a higher symmetry. If 



!) W. Voigt, Abh. der Ges. der Wiss. Gottingen, 36, (1890); Phys. Zeits., 17, 

 287, 307, (1916); 18, 58, (1917). 



2 ) Th. Liebisch, Grundriss der physikalischen Krystallographie, (1896), p. 

 177183. 



