9 



293 



where the inlegrntion iiiusi obviously be taken once up und down between the 

 hmils of oscillation 7' and 7" of cvei v r/. I^el us now assume that /V/,, ■ ■ Qs) (^an 

 be written in the I'orni of the sum of a finite number of products of functions, 

 which each depend on one of the (/'s only: 



tVh- ■ ■ <ls) ^ I\r^(l0hr(q>) fsr((Is). (16) 



;• 



Then it is easily seen that the value of the coefficient j^iven by (15), will be 

 equal to the sum of a finite number of products 



^ dhv'hr ■ ■ <lKr: 



(17) 



wliere </»,•,- is a definite integral of the fori 



d Si 



(Ih = \çf(<7,)e "^T'-^/fe dqi. (18) 



The character of these integrals may be brought out clearly by eifecling the 

 Ira nsformation 



An oscillation of f/,- up and down between its limits f/, and 7, corresponds to an 

 increase of ^, by 2;r. Further the functions will be periodic in tpi with period 

 2-, unless k i, in which case we have obviously 



= periodic function of </', (period 2;:). 



The integral (1(S) may therefore be written, denoting by /^„,P, ,.. /-*,, a set of 

 periodic functions of <p with period 2 r, in the form 



It is possible to express the coefficients C in the simple form given by (17), 

 only if the function f{q^^, . . . 7.,) that we want to expand in a trigonometric series 

 can be written in the torm (16). Now in the (|uantum theory a series expansion 

 of the rectangular Cartesian coordinates which describe the positions of the particles 

 of the system in space is asked for, and it might be of interest to investigate 

 whether these latter coordinates always may be expressed in terms of the coordinates 

 7i , . 7s, in which separation of variables was obtained, by a lormula of the form 

 (1(V). If the set of coordinates 7,, ... 7» belongs to the well known class of ..elliptical 

 coordinates", it is at once seen from the genera! formulæ holding for this kind 



1). K. 1). Vldensk. Sclsk. Skr.. iiaturvidensU. oi; iiK.lheni. M.I H Ha-kke. III. i. 38 



