94 



suitable form of / for this purpose. If this really be the case, the 

 figure / must be of such a nature that, if in P three coordinate-axes 

 OX, OY, and OZ are taken parallel to those of the whole system under 

 investigation, the general coordinates of P with respect to the coordi- 

 nate-axes of the system will appear also in the analytical expressions 

 which determine the image / with respect to the axes X, Y and Z. 



A velocity, a force, etc., may thus be represented by a heteropolar 

 vector (arrow) in P, determined by direction and magnitude, and 

 having the symmetry C . But there are a number of physical pheno- 

 mena for which the choice of the "image" in every point P of the 

 system is not so simple : in the case of a liquid endowed with optical 

 activity e.g., the symmetry may be represented by a portion of a 

 small screw, with its axis varying in direction from one point to the 

 other. 



In this connection it may be remarked, that a body or a medium 

 in which all kinds of physical phenomena can take place, may be 

 considered with respect to its symmetry-properties from several 

 standpoints: we can speak of the symmetry of the medium itself, 

 in so far as that symmetry is attributed to it from the point of view 

 of its molecular arrangement (crystals e.g.), or with respect to the 

 whole complex of phenomena observed in it ; or we can more particu- 

 larly draw attention to the symmetry of the medium with respect 

 to a certain group of phenomena, or finally with respect to a special 

 phenomenon only. In a similar way we "can speak of the symmetry 

 of a group of phenomena, or of the symmetry of a special phenomenon. 

 In all such cases we must know either the effects produced, or the 

 causes which produce these effects ; and the above mentioned image / 

 must represent certain elements of symmetry which we attribute 

 to the effects, or to the causes, if it should be really considered to 

 fulfil the condition, that it gives a complete description of the pecu- 

 liarities of the phenomenon under investigation. If this be the case, 

 the image / is indeed suited to its purpose ; and then it will be possible 

 for us to bring the considerations on symmetry-properties, as devel- 

 oped in the previous chapters, into the range of the phenomena 

 investigated. 



10. With respect to the connection of the symmetry of causes 

 and of the effects produced by them, we can now conclude from the 

 facts observed up till now, that differences of symmetry in the causes, 

 or in the special circumstances, can generally be manifested also 

 in the effects produced, but that this is not absolutely necessary in 



