1301 



~-^= 2. ai j kj + JS bij loj, 



d^ J J 



dki -vil X' i' 



— - = — 2. a ij hj — ^ b ij Vj, 



ar , j 



(23) 



dvi 



— = ^ dij kj -\- 2 eij loj, 



dr j j 



-- = — 2 d'ij hj — 2eij Vj. 

 dr 



The right-haiid members have no constant term. For hi and v, 

 these terras are zero in consequence of the conditions of symmetry 

 (16), since thej contain only sines. For ki and to, they are zero 

 by the conditions (18) and (19). 



The equations (23) are satisfied by 



hi =z 2 ciq Bq COS (fq, r» = 2 c'" fq cos rpq, 



ki = 2 c'ig Eg sin <pq, co,- = ^ cj.' Sq sin (fq. ,^ . . . (24) 



(fq = ^qr 4- 'WqO. ' 



Substituting (24) in (23) we find for dq, c'.^, c.'^, c|.^' and ^q the 



conditions 



Ciq ^q-\- 2 aij Cjq + 2 bij c" — 0, 



j j ^^ 



C'iq^q + 2 a'ij Cjq ^ 2 b'ij cj^ = 0, 



•^ ^ } . . . . (25) 



<•" P^9 + -^ dij C'j'j + ^ ^ij «•„ = ^. 

 y j j Jf 



The condition that it shall be possible to determine Ciq, c'iq .... 

 from these equations is that their determinant is zero. This gives 

 an equation of the sixteentli degree in /3^. To each root i^q belongs 

 a set dq .... There are however not 16 different values of /^^. To 

 begin with it is evident that, if we change ^q to — cpq, and conse- 

 quently i3q to — ^q, and if at the same time we replace c\q and c"iq 

 by — c'iq and — c"iq, the .equations (25) are still satisfied, and (24) is 

 not affected at all. It follows that if ^q is a root, then also — ^q is 

 a root. 



Further there are six roots (3 = 0. Each term in the equations (24), 

 i. e. each root /?, represents an oscillation of the true motion with 



