soAfE co^^l^^t^'Ol^'.^KY .iih-.i.whs ix /7/).s7c.s-/.v (n\ 



all iin»'rsi'-s<|ii.iri- I'li-ld .mil ,m imiform ina^;nolif tk-ld -separation of 

 \arial>li's is possiMo. I'Or these cases, llierefore, the conditions (70) 

 and (71) .ire apphcable, and have meaning;. 



There is t)nc other important case in wliicli it is jjossihle so to select 

 tiie variables that se|)aration can be effected. This is the case of an 

 electron niovini; according; to the laws of Newtonian mechanics in a 

 tUld compounded of an inverse-square field anfl an uniform electric 

 tield. Althoujjh the motion is three-dimensional, and three coordi- 

 nates are required and sutTue to determine it, these three coordinates 

 ma>- not be chosen at rantlom; and the three obvious ones would be 

 worthless for our purpose. If we should choose the polar coordinates 

 r. 6, and ^ employed in formulating the ecjuations (71), we should 

 tind that the momenta p,, />„ and /)^ do not depend each exclusively 

 uj)on the \ariable to which it corresponds. The procedure to be 

 followed is ainthing but ob\ious; but Jacobi found that if paraboloidal 

 coordinates are usetl instead of ]M)lar, separation of variables can be 

 effectetl. One must visualize two families of coaxial and confocal 

 paralioloids, their common focus at the nucleus, their noses pointing 

 in opposite directions along their common axis which is the line drawn 

 through the nucleus parallel to the electric field. The position of 

 any point through which the electron may pass is given by the para- 

 meters J and T) of the two paraboloids which intersect at that point, 

 and by an angle <t> defining its azimuth in the plane normal to the axis, 

 ciuite like the angle ^ of a system of polar coordinates. When the 

 motion of the electron is expressed in terms of these coordinates, the 

 corresponding momenta />£ and />, depend only upon ^ and -q respec- 

 tively and p^ is constant; hence the integrals taken over cycles of 

 f, rj, and respectively, on the right-hand sides of the equations, 



) /Jfrf? = w,/;, I pr,({n = nji J p^d<f> = II Ji (72) 



have definite meanings, and the e(|iiations themselves define particu- 

 lar orbits. Epstein determined the orbits defined by these ecjua- 

 tions, and calculated their energy-values. These agreed well with 

 the energy-values of the stationary states of h>drogen in an electric 

 field, inferred from its spectrum. This is the fifth of the striking 

 numerical agreements upon which the credit of Bohr's atom-model 

 chiefly depends '^. 



"Set- Epstein's article (.Imm. <l. Phys. SO, pp. 489-.S20; 1016), or the more per- 

 spicuous account by SommerfeUI. in which it is stated that the pattern of the com- 

 ponents into which the first four lines of the Balnier scries are resolved by the electric 

 field agrees with the predictions so far as the number and relative spacings of the 

 components are concerned; while to attain agreement in regard to the absolute 

 spacings, it is necessary only to assume that Stark's estimate of the field was 3','c 

 in error, which is (|uite easy to accept. 



