388 J. H. ROHRS, ESQ. ON THE 



• 



Consequently as far as terms involving — the chain is not quite fastened at its ends, in 



c 

 order to render it so, we must return to our original equation, 



sd?V ldV 



u = + — — , 



c ds c as 



where — — = "E.A n sin h 2.jB„ sin , 



ds 2 a a 



— — =Zi.B n — cos n ir, when s = a ; 

 aV a 



.-. = 7T (P, - 2 # 2 + S.B 3 ) + ^ - A 2 + A 3 . 



d?r 



Again, L x + — - = 0, s = a, ; 



.-. X, - (S x - 2B 2 + 3S 3 ) - = 0. 



a 



And x(5i- 2B 2 + 3B 3 ) = .14A7' = .14^ 3 ; .-. .4, - ^ + 1.144, = 0; 



d 2 , 2 8a d* „ tt s . 8a d 2 tt ,_ 



df <r ir 2 dv 4a 2 tt dr a 



&c. = &c. 



&c. = &c. 



d 8 7T 2 



Whence — (9 A 2 + AJ - - eg — (81 A % + ^,), 



ar 4a 



— (^ - 25 ^ 3 ) = - eg — (^ - 625 4,)- 



Whence A 3 = h cos v/24. 13. c# — £, 



2a 



A,= .165 h(T), 

 A y =- .975 h(T), 

 writing T as before for the periodic function. Maximum disturbance at centre from 

 L + ~2.A U series (taking in the two small terms A 4 , A s ) is .92 A'. 

 From "2.B„ series is .03 A' ; 



.*. .95 A' is the maximum disturbance nearly. 

 The maximum increase of tension at the ends is 124. h'g T nearly. 



If we compare this with the results obtained by neglecting the play " .01 8 A' we shall 

 find ,96k'* nearly for the maximum disturbance, and 122. h'gT for the maximum increase of 



d 3 V s* 

 tension ; the difference therefore is imperceptible. We might include the term — — - , al- 



u 6' 2 c 



* Including the part derived from 2. B„ series. 



