354 Astronomical and Nautical Collections. 



curve, which is expressed, for a small arc of vibration, by one- 

 eighth of the verse sine, the whole time of the vibration being 

 unity, or, for the arc A, since the verse sine of 1° is .000152, by 

 very nearly .000019 A'; and the fluxion of the time being dt, 

 that of the circular excess will be as A'dt = (B — Cf dt = 



C« dt — 2BC dt + B^dt: now A == 1 + —tOT:=\ + pt, 



B q 



putting p = — , and B = 6 ,and / = — 



^ ^ ^ q i +pt J ^ + Pt p 



hi (1 + pt), consequently the fluent of the second term is 



- 2C — hi (1 + pt) — - 2Cq hi A ; that of the third, or 



p B 



dt, being, when corrected, — . — i- ^t'^ 



il +pty ' &' p I +pt I + pt 



= bH — ~ bBt ; so that the whole circular excess will be- 

 6 



come .000019^ (C^ - 2C A hi (1 + pt) + ^ ^^ \ or 

 ^ pt I + ptj 



.000019K- -41 - 1.28 -IhlA + iB) = .00001 (1.96B 

 t B 



— 2.432 ^hl—- + .779.) Taking for example. Captain 



t B 



Kater's first register of experiments, in which a = 1°.38, and 



e h Q, 02 



A .92, when t was — , so that -£- being ±r± = 1.2949 = 

 2 B l.oo 



1 + _ < = 1 + _: — ; we must here make y rr ' — 



.2949 



17.124, and i- 6.850, and hi A being = .7031 -.4447 = 

 t B 



.2584, the whole is .00001 (5.987 - 4.304 + .779) t = 

 .00002462, or 2.12 in 86050 vibrations; which agrees exactly 

 with Captain Kater's computation from the separate arcs ob- 

 served. 

 If we adopted the Newtonian hypothesis of a resistance raea- 



