122 
PRESTON. 
and after integration 
1 I* A 
n u 0 + 4 u, 
0 + ( 
&C 
2 ) 1.2 ’ 
The data permit us to calculate the successive differences of 
u 0 up to d n u 0 . When the curve is divided into six parts, 
Weddle noticed that by changing the coefficient of the sixth 
difference by the y^-g-th part of itself, all the coefficients were 
rendered a common multiple; and since the sixth differences 
are from the nature of the approximation considered small, 
this introduces a very slight error, and we have 
^ u x dx — io [_ w o + u 2 + w 4 4- -f* 5 (wj + u^) -f- 6 u 5 
o 
where dx of the original equation is replaced by h. To 
apply this to the corrections for arc in pendulum work, the 
arc is read at the beginning, end, and at five equidistant 
intermediate points, dividing the swing into six equal parts. 
These are the values of u. The value of li being converted 
into seconds of time and the whole quantity being divided 
by 16, we get the sum of all the arc corrections. For the 
same swing already calculated, the correction for one oscilla¬ 
tion is 0 S .00000176. 
IY. The arc is read off when it reaches certain divisions 
of the amplitude scale—the intervals of time to the nearest 
tenth of a minute being also noted. These arcs are squared 
and divided by 16. Then taking successive means, we have 
arc corrections that apply quite approximately to the whole 
partial period between the different consecutive divisions of 
the scale. Weighting these in proportion to the length of 
the interval and taking the mean gives the final arc correc¬ 
tion applicable to the mean oscillation between the initial 
and final arcs. The introduction of weights depending on 
the time is nothing more than giving to each interval its 
requisite number of corrections, each of which is a mean 
between the initial and final arcs for that part of the curve 
of descent. This method gives a correction of 0.00000172 
