Mr. Gilbert on the Vibrations of Heavy Bodies. 97 



In the case of a mercurial pendulum, these quantities must be 

 reduced to three-fifths (.6) of their magnitudes in the table. 



It is then ascertained 



That the time of free descent down a given line, 

 The time of descent down the whole or any part of a cycloi- 

 dical arc of the same height by the semi-vibration of pendulum 

 having a suspension twice as long; 



And the time of a semi-vibration by the same pendulum in a 

 circular arc, will be, in the proportions to each other of 

 Unity, 

 Unity X quadrantial arc, 



Unity X quadrantial arc x (1 + — ). — 4- — ) + 



— + . — &c. &c.) 



48 63 385* 6^ 



Or substituting for a, the chord of semi-vibration in the vi- 

 bratory circle, which is in magnitude double to a, but in refer- 

 ence to its own radius taken as unity, will be one half of a, and 

 writing its values for b ; the series becomes 



1 + J_ V -£! 4- i- ^ c^ isvcs 105 y c^ 



Y/ '2^ y/ '2* 48 / 26 384 / 28&c. 



If s the sine of — the arc of semivibration be substituted, 



2 



the series becomes 



1 -H — ] . s- + _ I s* + — I s^ + ) s^ &c. 



2 / 8 / 48 y 384 / 



or if u = the verse sine, the series becomes 



2/2 8/22 48 y 23 384 ) '2* &c. 



Thus far the investigations are strictly correct ; but for all 



practical cases of vibration in small arcs, the two first terms of 



the series need alone be regarded, and the second only in its 



3 2 a* 



first power, since the third term — \ — - does not amount 



to one second in 24 hours till the arc of semi-vibration reaches 

 10° 5' ;northe square of the second term till the arc is 13° 24'. 



