194 F. A. P. Barnard on the Pendulum. Fa 
and — , or during the time found by integrating the expression 
already given for d ¢ between the same limits. Then, — 
_Img ff 1.62, 1.885 | 1.8.667 2) ae 
ciel Ee E+ 2.3a8* 2.4,508 ' 3.4,6.70” ta): 0) 
Or, putting S for the sum of the series within the brackets, ' 
2mgS , i: 
mg (t—t)= —" , which must equal A. 
Therefore, 
2mgS  g*la? 
; ~! =i and 4 Sk? m = gla? 2, 
Whence, 
_ wa glara _ Mgl?a2x ae 
Tal, Choe eee oN 
The foregoing series rapidly converges, and if 6 = ‘707s, its 
sum is¢7. Putting «, the length of the pendulum arm, meas 
ured at right angles to the pendulum rod from the centre of mo ; 
tion, = 3 inches, and employing for the other — the values 
heretofore given, we shall obtain for m and w the numerical 
values, 
m = ‘000001597. 
w = 2°914 grains, or 3 gr. nearly. 
Returning to the expressions (1) and (2), with the value of m 
: thus determined, and still employing for @ the value |S ri re 
4 T= 2X 86400 X 000001597 X 1384 _ 9.479 seoonds 
3°14159 X 035001597 : 
Whence it appears that this pendulum, in order to beat oe 
onds, must he about three one thousandths of an inch longer» 
one entirely free. at 
In order to investigate the liability of this pendulum tose 
of rate, we must observe that, at a constant temperature, fi 
impossible that there should be a change of rate without a pe : 
of the are of vibration; and further, that there is no ago 
operation to change the arc, except variations of density 1 hee 
Tn expression (5) we observe that «? varies as k? - but itis evi0 Or, 
that &? varies inversely as the density of the atmosphere. © 
putting D for the density, ; 
1 
a2 2 pers 
ok? aw D 
dD 
2adunrhdka——. 
