166 Mr. A. Stephenson on the Frequency Ranges of 



according to which end of the range n approaches. On the 

 former alternative one particular solution is given by 



90 



x = i Ar cos rnt 



-00 



00 



= A + 2 1 A r cos rnt. 



i 



and the other by 



00 1 °° 



x— Lt e=0 {t 2 A r cos rnt % A r sin rnt} 



— 00 <?— 00 



= Lt c=0 {t(Ao + 2 I A r cos rnt) -22 An^Arr s in mi}. 

 i i c 



Thus the solutions can be expressed by convergent series 

 of the form 



00 



#= 2 B r cosrwi 



and 



#= £ 2 B r cos m£ -*■ 2 C r sin rw£. 

 o i 



By direct substitution 



B + « 1 B 1 + a 2 B 2 + « 3 B 3 + ... =0, 



2« 1 B +]l-(? i / At )24-a 2 }B 1 +(a 1 + «3)B 2 +(« 2 + «4)B 3 + ... =0, 

 2a 2 B -4-(a 1 + « 8 )B 1 4 {l-(2w,/M) 2 + «4}B 2 + (a ] + «5)B 3 + > ..=0 ? 

 2«8Bo + («, + « 4 )Bi + (« 1 + «fi)B 2 +{l-(3n/^)- + a 8 }B 8 +...=-0; 



also 



J 1- (^) 2 -« 2 [ Q + ( ai _a 3 )C 2 + (« 2 -« 4 )C 3 + ... ==2n.B l5 

 («i-« 3 )C 1 +{l~(2n/A0 2 -«4}C 2 -r-(« 1 -a 5 )C 3 ... = 2n.2B„ 

 (a 2 - a4 )C 1 4-(a 1 -^)0 2 +^l-(3^) 2 -« 6 [C 3 +...=2^.3B 3 . 



The eliminant of the former set gives 



2«i 



-f« 2 a ]+«3 



!*3 



«2 + «< 



2 " I_(;;): 



2a 2 «i + a 3 1 — 4/ -I +a 4 «i + a s 



2a 3 a 2 + « 4 «i + a 5 1 — 9 J -J + 



*6 



= 0, . (2) 



