System about a Configuration of Equilibrium. 453 



But if 



c 2 — n 2 a 2 = 0, 



the inference that Ki = does not follow ; and if 



c 2 — 4n 2 a 2 = 0, 



the terms in cos 2nt in (10), (11) assume infinite values. 

 Accordingly these two cases demand further consideration. 

 We will commence with that where 



c 2 — n 2 a 2 = 0, 

 that is, where both modes of infinitesimal vibration have 

 the same frequency. 



We must now discard the supposition that </> 2 = approxi- 

 mately and be prepared to allow K l3 as well as H 1? to be 

 quantities of the first order of smallness. The other coeffi- 

 cients in (6), (7) are still of the second order at least. Sub- 

 stituting in (4), (5) and retaining only terms not above the 

 second order, we get 



t\ H + (c : — ?i 2 <z 1 )H 1 cos nt + (ci — 4ra 2 a 1 )H 2 cos 2nt 4- . . . 



+ 3 7l H^ 2 cos' 2 nt -f 27 2 H^ K x cos 2 nt + y 3 Kx 2 cos 2 nt = 0, 

 c 2 K + (c 2 — n 2 a 2 )K 1 cosni-f(c 2 — 4w 2 a 2 )K 2 cos 2rc£ + . . . 



-f 3^ Kx 2 cos 2 nt + 2y 3 Hj K x cos 2 nt -f y 2 Hi 2 cos 2 nt = ; 



whence 



c 1 H +i(3 7l H 1 2 +2 72 H 1 K 1 + 73 K 1 2 )=0, . (12) 



^K + K37,K 1 2 + 2 73 H 1 K 1 + 7 2 H 1 2 ) = 0, . (13) 



(c 1 -4n 2 a 1 )H 2 + K3yiH 1 2 + 2 72 H 1 K 1 + 73 K 1 2 ) = 0, . (14) 



(c 2 - 4n 2 a 2 )K 2 + J (3 74 Ki 2 + 2 73 H 1 K 1 + 72 H^) = 0. . (15) 



Also 



H 3 , &c, K 3 , &c. = 0. 



These equations, arising from the terms independent of t and 

 proportional to cos 2nt, cos Snt, &c, determine H , K , H 2 , 

 K 2 , &c. when H 1? K l5 and n are known. The term in cosn£ 

 gives further 



H^Ci — >i 2 a 1 ) = 0, Kx(c2 — n 2 a 2 ) = 0. 



Thus when 



n 2 = Cj/a! = c 2 /a 2 , (16) 



Ki as well as Hj is an arbitrary quantity of the first order. 

 And this completes the solution to the second approxi- 

 mation. 



