OSCILLATIONS OF A SUSPENSION CHAIN. 387 



2s d 2 , dV 



6. To include the term — -— (V), let us assume for — — an additional series of 



c dr ds 



„ . irsn . . ... 



sines 2 . /»,, sin representing a function which vanishes when s = and a = a. 



a 



In — write for s — - sin — , which will be near enough for our purpose without taking 

 C* ir* 2 a o r r ft 



in more terms of the series 



d i V (tfA-L its 2a d* 3vs 2a d 2 Sirs 2a\ 



— r = — —rr cos — x — + — - A 2 cos x — + — A, cos x — 



df \df 2a 7T dt 2 2a 3tt dt* 3 2a 5tJ 



28 d* V 1 /d 2 ^, . tts_ ld*A 2 . 7ts\ 



7 d7" = " ^ \lt sm ~a~~3~d! r 8,n ~a~) 



1 /ld 2 ^ . 2tts 1<PA 3 . 2tts\ 



x — =-=■ sin — — sm 1 



7T 3 \3 df a 5 dt 2 a } 



1 /l d 2 ^ . 3irs\ . . , 16a 2 



? 7 "T^~ sm » remembering that — — = 1, 



it 6 \5 dt a J c 2 



d 2 B l w* „ 1 «f* . 1 d 2 , 



dr a'' 7r 3 dr Srrdr 



d 2 „ 4tt 2 _ l d 2 , id 2 , 



— - A] = — cff Z? 2 A, + — - — A 3 



df ' S a 2 2 3 7r 3 d^ 2 5tt 3 d* 2 3 



— - B 3 = — eg — B 3 At, 



dt 2 s a 2 3 57r 3 d< 2 3 ' 



also A, = — A cos \Zcg.24-.2 — t, nearly, 



3 5 s 2a 



A% A 3 3h 



— = — - . T, (writing T for the periodic function), 



A * A m 



— m - T. 



5 5 



Retaining only the periodic T, and putting the coefficients of cos y/cg — t &c. = 0, we have 



#, = — (1.07 hT) nearly, 



IT 



# 2 =— , (.45A7 7 ) . . . 



B 3 = — 3 (A2hT) 



The value of v when « = a derived from this series is .018A' nearly at its maximum. 

 Vol. IX. Paet III. 50 



