Oscillations of Chains. 741 



We find, after simplification, that 



tan(fcr-|+*)= tan (*-?+„); 



4: 4 



or 



(X-l)^ = ^7T + 9 7 -6'. [/l= 1, 2, 3 ... ] 



Writing y for — , we get from (4) 

 oz 



y 3V __ 6834 ?/ 5 

 X f X 3 X 5 



tantf=-^ f ^f-_^^ + 



and making use of Gregory's series, 



0- _i_ 100 34336 



8^ + 3\ 3 (8;) 3 "~5AJ>(8^ 5 + W 



Similarly _ 15 180_ 47520 



V ~S~;~(^f~~ (8;) 5 •■ W 



Since nrr 1 



we find, on substituting the values of tj and #, and writing /3 

 tor 



A-l' 



15X + 1 540X. 3 4-100 _ 237600X^-343 36 



"~^ + 8\(\--l)z~~3\ 3 ,8 3 . (A.-l> a 5A/\8 5 (\-l> 5 ■"' 



an equation of the form 



. = />+£ + & + £+■■., 



where 15X + 1 _ _ 135\ 3 + 25 



i '~8\(X-l/ ^ 384\ 3 (A-1) 



and 7425X 5 - 1073 



5120A 5 (A-1) * 

 By Lagrange's theorem 



Close approximations to the earlier roots are not given by 



