21 



305 



^n.T, = , [aiJr^{Ta^)J'T^(vao_) — a^Jr^(7a^)JT^\ra.-.\) . (59) 

 As regards the term .4,, „ in the expansion (ô4) for r, we have ohviously 



/(/i — /a) H- ^ \ \ (Tj cos ç'' — a, COS ( 1 + <T, COS ^ -h 'To cos d(/i 



= xl{l, ~ ^2) + -'^ ) - 2 '^^'^^ ~ 



The expansion for z in a trigonometric series therefore assumes the form 

 Z = |;f/(/, - /,) + xPl^\l^rJ.^Jr,iTa,)JT,(Ta^) a^J'r,{z(T,)Jz,{Ta^) j e2;r,(r, + r.«,), (ßQ) 



where the summation is to he extended over all positive and negative entire values 

 of Tj and r^, with exception of the combination = 0, 73 = 0. For the combina- 

 tions for which r = + ^2 = the expression for the coefficients becomes unde- 

 fined, but by introducing r = in (57) it is easily seen that the coefficients in 

 question are equal to zero. 



In order now to find the trigonometric series representing the displacement 

 of the electron in the direction of the x-axis and of the y-axis, we might follow 

 the procedure indicated in § 1, but the calculations may be made shorter, just as 

 in § 2, if we observe that the 2-axis is an axis of symmetry of the system, as a 

 consequence of which the expansion for x -{- iy will only contain terms of the type 



Mm, + L, cost'' °^ (M, + + £,cos^ 



{(M,+ ^2)cos| +//fsin| j 

 Tl;^ = - nog (M^qTL^) iM,-\- L,œs 



the equations (51) (b) and (c) give 



^, ^ , {(Ml ^Li)cos|-+i/fsin |-}' + Ljcos ^ -f lÄsin ^ 



2 - (u;, — ) = - 2 i- 2 ~ ^ 2 '"^ \M,^ L\)JM^Q)XM, + L, cos {M, ^ L, cos y ) 



so that, making use of (43 1, we have 



(.r r n/)e-'^'^"«^"'"' -=- \/ ~rj e'V + -^i^>"t - <•■' 



