192 F. A. P. Barnard on the Pendulum. 
Developing these arcs in terms of their sines, according to the 
ordinary series, taking their difference, and expr by 4r the 
difference between z and the time of vibration of a free pendulum, 
ar=— FF @+m)=(0—m)  1[8-+-m)>—(—ms) 
NgL (+m) t 2.3 (am) * a 
1,3 [(P-+-m)* —(8 — m)5] ; 
+ 2.4.5 (a-+-m)§ ke | 
which becomes, if we neglect insignificant terms, 
e __ 2m Z 2 Bt gs 
aa or; [}+o V3 det + 34.600? &] () 
If ts as for the free pendulum beating seconds, then, put 
ting 47’ for the total acceleration of the clock in making the 
number of beats made by the seconds pendulum in a day, 
calling the entire remaining value of the second member of the 
last equation S, Re 
47T= ee od (2) 
_ The terms containing 7 having disappeared from these expres 
sions, it would seem that the resistance of the air does not ailect 
the time of vibration. These terms, however, have not been 
eliminated, but only neglected, in consequence of ya 
“ 
of the air opposes gravity during the descent of the pendulum, 
ent. 
The value of the foregoing series depends upon the impellng 
column of air of equal base and weight would be about 1 de 
feet in height, and the velocity with which a fluid of this alttade 
