789 



2a Ru 



( Oil 2a . a)l\ 



I cos sin — 1 . 



V 2a loy 2a) 



The identity of these frequency-equations can l)e easily shown. 

 Fut — - — i = k, then (6a) takes the form 



8 ,_ 1 8 nJü) 



1 —k. — 2 L/,.._v i_ _ o 



The sum of inverse squares of odd numbers is — . Further. 



8 



/is^ =z — -—, therefore the tirst member amounts to 



k ^ a' 1 

 1 \-Sk — ^ = 0. 



CO Po) Hs^ — tO^ 



For tg z we have 



2z 



tgz = -:E 



1 



■-(?)' 



where s is again an odd number, therefore we obtain 



k 2a 0)1 



l-- + -—ktg-=0 (14) 



tt> (o I 2a 



The equation (6) takes the form 



Oil f k 2a o)l\ 



aicos-(l-~ + —ktg-] = i) .... (15) 

 2a \ a% lio' 2a J 



The equations (14) and (15) have the same roots, for the vectors 



tol 

 oi and 6'(M' — do not contribute roots to (15). 



Having found the roots of (14), we can determine y. Each root 



sJTa 

 yields a Fourier series. In the case that (A* = oo), i^ln — must be 



e 



combined witli one frequency only. For our case we have 



^ Ay, ico/t . sjTx 



y ■=. 2 s ^f. ; . e sin 



s(ns^ — fJi"^) I 



1 simx Ay icDy.t 



2s - sin —— Sr ., e (16) 



s I ng' — lOy 



iiOyt 



The Fourier series which is the vector oï A e must be equal 



to the function which in § 2 appears as the vector of the same 

 exponential. This can be shown by dii'ect development. It is ai)pareut 

 that by a given frequency all the original normal coordinates are 



