System about a Configuration of Equilibrii 



455 



first order as well as H„ the remaining- coefficients being 

 still of: the second order. The substitution of (6), (7) in (4), 

 (5) then gives 



Ci B. -j- (c l —n 2 a 1 )B. 1 cos nt + (ci— 4/i 2 a 1 )H 2 cos %nt + , . . 



4- 87! H x 2 cos 2 rc£ + 27 2 H x K 2 cos ?i£ cos 2nt 



+ 7 3 K 2 2 cos 2 2^=:0, . . . (25) 



c 2 K + (c 2 — ?i 2 a 2 )KiCOsn£ + (c 2 — 4?z 2 a 2 )K 2 cos 2;^ + . . . 

 -f 374 K 2 2 cos 2 2nt + 2<y z H x K 2 cos nt cos 2nt 



+ y 2 H. 1 2 cos 2 ?it = 0. . . . {26) 



From the terms independent of t we get 



2c 1 H + 37 1 H 1 2 + 7 3 K 2 2 = 0, 2c 2 K + 72 H^ 2 + 3 74 K 2 2 = ; . 



from the terms in 3nt 



( Cl -9n 2 a 1 )H 3 H-7 2 H 1 K 2 = 0, (c 2 -9?i 2 a 2 )K z + y,IL 1 K 2 = 0; . 



from the terms in 4?i£ 



(ci — 16n 2 a!)H 4 + i7 3 K 2 2 = 0, (c 2 — 16?2 2 a 2 )K 4 = §7 4 K 3 2 ;. . 



while coefficients with higher suffixes than 4 vanish. 

 Further, from the terms in nt, 2nt 



(«!— n*a 1 )H 1 + 7 2 H 1 K 3 =0, (c 2 -n 2 a 2 )K 1 + 7 3 H 1 K 2 =0, . 



( Cl - 4n 2 aOH 2 + §7i H^ = 0, (e 2 - 4n 2 a 2 ) K 2 + i 72 H x 2 = 0. . 



These equations determine H , K , K l5 H 2 , H 3 , K 3 , H 4 , K 4 

 as functions of H x and K 2 of the second order, when n is 

 known. To find n and the ratio K^/Hi we have the first 

 equation of (30) and the second of (31). With regard to 

 (24) these may be written 



c i - 



)i 2 a, +y 2 K 2 = 0, 

 0; 



(33) 

 (34) 



of which the first may be considered to determine n. Elimi- 

 nating (pi — n 2 ai), we get 



Ks/Ei = ±y/(e 1 /2e 3 ) (35) 



This completes the solution to the second order of small 

 quantities. 



If V 3 = 0, the above solution reduces itself to that of the 

 first approximation. In this case, especially if V is an even 



(27) 

 (28) 

 (29) 



(30) 

 (31) 



