100 BELL SYSTEM TECH NIC A L JOURNA L 



The quantities (^i , • • • , ;/3) are the cosines of the angles between the various 

 axes; thus A is the cosine of the angle between the axes Ox', and Ox; n^ the 

 cosine of the angle between Oz' and Oz, and so on. By solving the equations 

 (61) simultaneously, the coordinates .v, y, z can be expressed in terms of 

 .t', y' , z' by the equations. 



X = l,x' + t^' + t,z' 



y = mix' + Woy' + nviz' (62) 



z = nix' + n<iy' + r^z' . 



We can shorten the writing of equations (61) and (62) considerably by 

 changing the notation. Instead of x, y, z let us write .Ti , x? , Xz and in place 

 of x' , y' , z' we write X\ , X2 , Xs. We can now say that the coordinates with 

 respect to the first system are .Ti , where i may be 1, 2, 3 while those with 

 respect of the second system are Xj , where / = 1, 2 or 3. Then in (61) 

 each coordinate Xj is expressed as the sum of three terms depending on the 

 three x, . Each x, is associated with the cosine of the angle between the 

 direction of x, increasing and that of x, increasing. Let us denote this 

 cosine by c , y . Then we have for all values of j, 



3 



x'j = aijXi + a2jX2 + asjXs = ^ aijXi. (63) 



Conversely equation (62) can be written 



3 



Xi = XI ^•■y-'^y (64) 



y=i 



where the a ,; have the same value as in (63), for the same values of i and 7, 

 since in both cases the cosine of the angle is between the values of x; and x; 

 increasing. Such a set of three quantities involving a relation between two 

 coordinate systems is called a tensor of the first rank or a vector. 



We note that each of the equations (63), (64) is really a set of three equa- 

 tions. Where the suffix i or j appears on the left it is to be given in turn 

 all the values 1, 2, 3 and the resulting equation is one of the set. In each 

 such equation the right side is the sum of three terms obtained by giving j 

 or / the values 1, 2, 3 in turn and adding. Whenever such a summation 

 occurs a suffix is repeated in the expression for the general term as dijXj . 

 We make it a regular convention that whenever a suffiix is repeated it is 

 to be given all possible values and that the terms arc to be added for all. 

 Then (63) can be written simply as 



x^ = a,;X,- 



the summation being automatically understood by the convention. 



