Perpend IV aliw it ij ui a Parallelepipedal Sysieni. ] J) 



lines and planes passing through a fixed point of the systeni 

 taken as origin. 



3. AVhenever a line of the system is perpendicular to a 

 plane of the system, the .system has a certain " symmetry of 

 aspect" with regard to that plane. Let O be the plane, and 

 let be any point of the system lying in it. The planes and 

 lines of the system which pass through are symmetrically 

 distributed with regard to Cl ; but the points of the system are 

 not (in general) symmetrically distributed with regard to fl i 

 thus, if OP is any line of the system not lying in the plane O, 

 and if OQ is the reflection of O with regard to the plane O, 

 OQ is a line of the system as well as OP, but the points of the 

 system which lie on OQ are not (in general) the reflections of 

 the points of the system which lie on OP. Hence, while the 

 points of the system are not themselves symmetrically distri- 

 buted with regard to fl, the directions in which they would be 

 "s^iewed by an eye situated at are symmetrically distributed ; 

 and this is what we intend to express by saying that the sys- 

 tem has a " symmetry of aspect " with regard to the plane fl. 



As we shall have no occasion in what follows to consider 

 planes of absolute symmetry, we shall for the sake of brevity 

 use the word symmetry in the sense of " symmetry of aspect." 

 Thus any line and any plane of the system which are at right 

 angles to one another are an axis and a plane of symmetry. 



4. The cases of symmetry, as thus defined, which can present 

 themselves in a parallelepipedal system are four in number. 

 There is (1) the case of simple symmetry, when there is only 

 one axis and one plane of symmetry ; and there are three cases 

 of triple symmetry, which may be characterized as (2) the 

 ellipsoidal, (3) the spheroidal, and (4) the spherical. In an 

 ellipsoidal system there are three mutually rectangular planes, 

 which are planes of symmetry; in a spheroidal system there 

 is one equatorial plane of symmetry, but every plane of the 

 system at right angles to this plane is also a plane of sym- 

 metry ; in a system having spherical symmetry every plane 

 of the system is a plane of symmetry, and every line of the 

 system an axis of symmetry. Two simple symmetries cannot 

 coexist without forming a triple symmetry, which is ellipsoidal 

 if the axis of one of the symmetries lies in the plane of the 

 other, but is spheroidal in every other case : three simple 

 s^mimetries form an ellipsoidal symmetry if the three axes are 

 at right angles to one another, a spheroidal symmetry if one 

 of the axes is at right angles to the plane of the other two 

 which are not at right angles to one another, a spherical sym- 

 metry in every other case. 



5. Adopting the notation of the classical treatise of Pro- 



C2 



