100 Mr. Davies Gilbert on the 



To determine the amount of circular excess in arcs con- 

 stantly diminishing from the effect of resistance, let a the larger 

 of two small arcs of descent which in any portion of time, con- 

 sidered as unity, diminishes to b, 



Let X = any portion of that time, 



y =; the arc of semivibration at that instant, 



m =: a modulus then x = "^^ and x = C + — 



y- y 



When a; = and w = a /. C = — — and x = — — — 

 ^ a y a 



And when x = 1 y = 6, consequently 1 == — - — — or ah 



, , ab ■, (lb ab 



=1 am — bm, whence m = and .r = — — 7 



a — b ay — by a- — ab 



whence is derived 



ab 

 y = 



a— by. x + b 



And '/'', or the circular excess, will be ■ 



16-^ ^^ {a-byx + by 



expressed in terms of a: and of known quantities. Then will 



_ " X X represent the fluxion of the variable 



16 {a—b .x-\-by 



excess of which the fluent is 



^ X - -1- X ___L + c 



16 a — b g^_i X X + 6 



Whena-=0, C=. "'^ 



16 X a—b 

 The whole fluent, therefore, 



a'^b _ a'¥ 1 



1 c V r Tc • - , , ! ", when x = 1 



16xa— 6 16 X a— 6 a—b y. x + b 



The fluent becomes 



— = — and this quantity multiplied by the number 



16xa-i 16 4 y 1 J 



of seconds observed between the two arcs of semi-vibration 

 a and 6, will give the whole circular excess in seconds. 



