6 BELL SYSTEM TECHNICAL JOURNAL 



and markings onto a plane through the center. Figures 1 to 32 is a set of 

 such diagrams. Fig. 23 for instance shows a six fold axis. Fig. 1 represents 

 a medium with no symmetry whatever. The cross represents a typical 

 vector property, the vector piercing the sphere above the projection sheet. 

 If the vector pierced below the sheet it would be marked with a circle. The 

 dashed circle of Fig. 23 indicates the boundary of the sphere without im- 

 plying it to be a plane of symmetry. The presence of six fold symmetry 

 requires the typical vector to be shown in six places. If an axis of two fold 

 symmetry is added at right angles to the six fold axis, it must appear six 

 times and the typical vector must now appear twelve times, six times above 

 and six times below the projection sheet. Continuing in this way we shall 

 find the self-consistent classes of symmetry to be the 32 shown in the dia- 

 grams. Often the symmetry of a crystal class is expressed by means of a 

 formula. A center of symmetry is symbolized by the letter C, a binary 

 axis by ^2, a trigonal axis by ^3, a ternary axis by ^4, a six fold axis by At, 

 a plane of reflection by P, and a combination rotation reflection by the 

 combination symbols A*4 or A*6- In this way the symmetry formula of 

 quartz for example, is 3^2" ^3- 



SECTION 4 



Matrix Al6ebra 



In the solution of problems of crystal physics we are involved in the 



handling of many sets of linear simultaneous equations. As the matrix 



algebra lessens the work involved in handling sets of linear simultaneous 



equations we turn now to a study of matrix algebra. 



Several independent variables .Ti, .T2 . . . .t„ are linearly related to several 

 other independent variables yi, 3^2 ... ym as 



yi = anXi + ai^Xi, -}-... Ci„a;„ 

 ji = a^xi -f . . . 



or briefly 



n 



yi = ^ aijXj I = 1, 2 • ' • m (4.1) 



3=1 



In most all such equa.tions as (4.1) the variable to be summed over appears 

 twice in the subscripts of one side. As a convention we agree to omit the 

 summation sign and sum wherever subscripts are repeated. 

 Thus: yi = aaXj is to be summed overj 

 again, if Xj = bjkZk the z's being a third set of variables we have: 



yi = dijhjkZk to be summed over 7 and k. 



