g PROFESSOR FORBES'S EXPERIMENTS ON 



riation into account) must be applied. The law of the diminution of arc, or the 

 factor representing the ratio of the arc of one vibration to that of the imme- 

 diately preceding one, will be at once deduced from observing after how many 

 vibrations the arc is halved. Let m be that number, then if r be the factor in 

 question, r"»=:^; whence r — '^>sj\, which is known; and therefore m, together 

 with the initial semi-arc of vibration, may be used as the arguments for entering 

 the following table of corrections : — * 



* Investigation, — Let » be the initial semi-arc of vibration (taken in parts of radius), and r the ratio 

 of its diminution by resistance in a single vibration. 



Then, for the 1st, 2d, 3d, 4th, wth vibration, 



The arcs will be, », »r, otr^, ctr^, ar^~^ 



And, by mechanics (Poisson, art. 184.), the times occupied by these vibrations will be (the time of an 

 infinitely small vibration being unity), 



And the mean duration of the n vibrations is. 



*2 



M = 1 -^ • — r -L 21 =1-1- — • -; 



^16 n ^ 16 (l—r')n 



Hence the mean duration of n vibrations, 



1— r 



From the 0th to the nth, is 1 + — .- ^, = 1 -f- ^-"^ . A 



^ 16 (1— r2«) ^ \4J 



10th (n -I- 10th), is = i + /*yr"*^'.A 



(because the initial arc instead of « is « r^°) 

 . 20th, (n + 20th), is • =14- (-VV" ^ "" . A 



60th, (« -f- 60th), is = l+f-Vr^'^KA 



And the mean value of these deviations, is 



The concluding factor may be called B ; and substituting the value of r from the text, or (-g)"? we 

 have 



■A. — , 2 \ ^ -^ / 20\ 



{\-{\y)n A^-{\r^ 



(the last factor being independent of n), and we have 



^ . J rr.. Observed Mean Time 

 Corrected Time = s ' 



,+Q.A.B. 



whence the Tables are computed. 



