120 Professor Airy on the Disttirbanees of Pendulums, 



— d- : and the increase caused by the force F (found in the same 



o "Pq 4 ^ 2 Fc 



manner) would be -^r- . Hence ^- «' = —r- , whence 

 ' n a 3 n a 



_, sA-n' a" 



F = — 



3 • c ' 



and the diminution in the time of vibration 



_ 4 k a A^a^ — c^ 

 ~ 3^ ■ ' c ■ 



The differential coefficient of this with respect to a is 



4k 2d-e 



S-n-' C^ 



■ c 



.« ■ 



Here then the vibrations are performed quicker in the large arcs 

 than in the small ones. 



If the force, instead of being constant from c to a and from 

 a to c, varied directly as the distance, putting, in Example 9, 

 M = ex, the time is diminished by 



2 /* ex- (X = CJ 



or by -4-^ {c 7^^^^ + «'(!- sin-' '-)] = -h W 'n - ^] ^ 



nearly, when c is small : which is almost independent of a. Her^ 

 then an alteration in the arc of vibration would scarcely affect 

 the clock's rate : but an alteration in the maintaining power would 

 affect it greatly. 



We may thus investigate the possibility of a law of resistance 

 that will make the vibrations isochronous, however the maintaining 

 power may vary. Suppose the force on the pallets constant in 



