EQUATIONS OF ELECTRON MOTION 173 



Now Hamilton's principle expresses a fact concerning the motion of a 

 particle which is, by its very nature, independent of the choice of coordinates. 

 Hence, the Lagrangian equations (6) hold in any coordinate system. How- 

 ever, the form of the function L depends upon the particular coordinate 

 sj'stem, and we must discuss the change of the form of the function resulting 

 from a transformation of the coordinate system. 



In accordance with the common practice in the tensor calculus, we shall 

 now denote the coordinates by the symbols .r\ x , x^, instead of by the 

 symbols Xi, X2, X3. 



In rectangular coordinates the differential distance ds between the points 

 (.v\ X-, x^) and {x^ + dx'^, x- + dx-, x^ + dx^) is given by the simple formula 



J52 = J.v-i' + dx^' + dx'' , 



but this is highly special; in general coordinates we have 



3 3 

 ds = 2-/ z2 Smn{x^, X , x) dx^ dx" , (37) 



m=l 7j=l 



where the g's are functions which depend upon the particular coordinate 

 system under consideration. It is understood that gmn = gnm ■ Hence- 

 forth, we shall write (37) in the form 



ds^ = imn{x\ x\ x')dx"'dx-, (38) 



and we shall observe this general rule throughout: When the same literal 

 index occurs twice in a term, once as a subscript and once as a superscript, 

 that term is understood to be summed for the three values of the index. 

 We now have the result 



v"^ = [ds/dt]^ = g,nn(x^, x^, x')x"^x'^, 



and 



WoC-(l - 7;2c-2)X/2 = fnoC''[l - C-^gmnX'^X"]"'-. 



The function V(x^, x^, x^, t) is a scalar. That is to say, when the coordi- 

 nate system is changed, the first three arguments of the function are replaced 

 by their expressions in terms of the new coordinates, and so we obtain a 

 function which is of a new analytical form, but which has the same value 

 as the original function at each point of space. 



Now we consider the expression 



AiX^ -\- A2X^ -\- Azx'. 



In rectangular coordinates this is the scalar product of the vectors (^i, 

 A2, A'^ and (a;^, x-, x'). The expression retains its form and interpretation 



