OSCILLATIONS OF A SUSPENSION CHAIN. 383 



/ . 2n + 1 irs\ 



2 [A x sin I . Then L = 0, 



d'V I . 2ra + 1 ir 2« + l vs \ 



-T-; = 2 [A n cos — - — . — , and L y = 0. 



ds* \ 2 a 2 a J 



At least when 2 A n is convergent as it is in all the cases I have considered. 



s 2 1 



Neglecting also — , the value of which lies between and — , and is therefore insignifi- 



cant, also u must = when s = ; .\ M x = ; 



d 2 (dV\ 2* d 2 [d?V\ 



We shall afterwards approximate to the solution of the entire equation, including 



d" 2 s 



— - -7 V, by substituting for V in that term its value found from (4), and working the equa- 



dt'~ (f 



tions over again, 



s can be expanded in a rapidly converging series, 



(8a(-l)" +1 . 2rc + l7rsl 



< —— sin > , 



\(2« + 1) 2 tt 2 2 ay 



between the limits s = and s = a, where a is the length of the half chain, or the value of s 

 from the lowest point, to the point of suspension. 



Substituting then in (4), and equating to zero the coefficients of like sines, we have 

 d i A l 2 8a d 3 „ tt 2 A 



dt* <r it' if 4a 2 



d z . 2 8a d* 9tt 2 



dF^-^-^d?^-- *-^^ (5) ' 



d 2 ^ 2 8a d 2 , 25. tt 2 



&c = &c. 



The condition that u may = when s = a renders it necessary that 



-4, - 4, + A 3 - &c. 



=0 (o). 



c 



If, however, we assume only A t - A 2 + A s = 0, and suppose the initial values of the 

 constants in the equations for A i , &c. to be =0, A i} A 5 , &c. will form an exceeding rapidly 



convergent series, the sum of which will be very small, so that — 6 may be neglected ; 



we shall thus have a vibration of three terms, and two types of vibration, we shall only retain 

 the type with the higher coefficient, and in this way we shall find 



A 3 *= h cos vcg - . 24.2 — t, 



A 2 = .15h cos \/cg. 24. 2 — t, 



2a 



