REDUCTION OF PENDULUM OBSERVATIONS. 
121 
constants may be determined graphically and approxi¬ 
mately by dividing by % the arc and thus reducing the equa¬ 
tion to that of a straight line. Then plotting the equation 
b + c <P m = n, 
b will be the value of the ordinate where the line cuts the 
axis of y, and c will be the tangent of the angle made with 
the axis of x. This method is so simple that a great many 
values of the arc may easily be introduced, and the straight 
line best satisfying all the values taken to determine the 
constants; but a few values at the beginning, middle, and 
end of the swing give all necessary accuracy. The con¬ 
stants being found and substituted in the definite integral 
equation 
a/ *=i [ Na p- ’og (1 + 7) - 7 ] 
a 
taken between the limiting values of the arc gives the 
quantity to be subtracted from the interval given by obser¬ 
vations to obtain that required to make the same number of 
oscillations in an infinitely small arc. The correction for 
one oscillation in the swing just referred to would be by 
this method s .00000173, differing from the preceding by 
only 4 units in the eighth place. 
Ill (Weddle).* 
Here the decrement of arc is plotted, and the curve 
divided into equal parts. Letting u represent the ordinates, 
and n the number of parts, we have to find the integral 
^u x dx. 
o 
This from the ordinary interpolation formula is equal to 
ft 7v 
A x dx + q~2° J* x (* — L 
dx , 
* Boole (Geo.) A Treatise on the calculus of finite differences. 12° Lon¬ 
don , 1860, pp. 38-39. 
