FOR DETERMINING THE MEAN DENSITY OF THE EARTH. 
115 
Table (continued). 
n. 
T n — 2 
Log . 
n 
n. 
T n—2 
Log 
n 
n. 
T n—2 
Log 
n 
w. 
r n—2 
Log 
n 
n. 
T n—2 
Log 
n 
488-0 
9*99821645 
489*3 
9-99822120 
490-6 
9*99822591 
491-9 
9-99823062 
493-2 
9-99823529 
•1 
1682 
•4 
2156 
*7 
2628 
492-0 
3098 
-3 
3565 
•2 
1718 
•5 
2193 
-8 
2664 
•1 
3134 
•4 
3601 
•3 
1755 
•6 
2229 
•9 
2701 
•2 
3170 
•5 
3637 
•4 
1791 
*7 
2265 
491-0 
2737 
-3 
3206 
-6 
3673 
•5 
1828 
•8 
2302 
•1 
2773 
-4 
3242 
*7 
3708 
•6 
1864 
*9 
2338 
•2 
2809 
•5 
3278 
-8 
3744 
•7 
1901 
490-0 
2374 
•3 
2845 
•6 
3314 
*9 
3779 
•8 
1937 
•1 
2410 
•4 
2881 
•7 
3350 
494-0 
3815 
•9 
1974 
•2 
2446 
-5 
2917 
-8 
3385 
489*0 
2010 
-3 
2483 
-6 
2953 
*9 
3421 
•1 
2047 
-4 
2519 
*7 
2989 
493-0 
3457 
•2 
2083 
•5 
2555 
•8 
3026 
-1 
3493 
1 
32. The number given by this table is the logarithm of the mean Rate of the De- 
tached Pendulum upon the Clock Pendulum under the actual circumstances of observa- 
tion. It is next required to investigate the correction to this logarithm depending on 
the extent of the arc of vibration ; one of the conditions of the data of the problem 
being, that the arc is observed only at the beginning and the end of the Swing. 
In the first place, to compute the correction to the logarithm, supposing the arc of 
vibration constant. Let the whole arc of vibration, as seen upon the scale of inches, 
be I. For the pendulum 1821, suppose the scale to be placed 1 inch behind the 
pendulum ; and for the pendulum 8, r8 inch behind the pendulum. And suppose 
the distance of the object-glass of the observing telescope to be 100 inches. Then 
the real whole arc of vibration is Ix for pendulum 1821 and Ix fof pen- 
dulum 8. The lengths of the two pendulums, from the knife-edge to the indicating 
point of the tail, are respectively 60'7 and 60*2 inches. Hence the proportion of the 
real whole arc of vibration to the length of the pendulum is Ix fb/x 60-7 
dulum 1821 and Ix ioi-8^66-2 pendulum 8 ; in which expressions the factors of 
I are sensibly the same. Call this proportion C. Then the number of vibrations 
observed is to be multiplied by 1+ ^ or l + Px^y ( ibi x 60- 7 ) ' instance 
therefore we require the logarithm of l + PXgy ( loi x60-7 y values of I not 
exceeding 2’5. The following Table contains the numbers required, in units of the 
last figures of 8-figure logarithms. 
I. 
Log. 
I. 
Log. 
I. 
Log. 
I. 
Log. 
I. 
Log. 
0-1 
2 
0-6 
65 
1-1 
218 
1-6 
462 
2*1 
796 
0-2 
7 
0-7 
88 
1-2 
260 
1*7 
522 
2-2 
874 
0-3 
16 
0-8 
116 
1-3 
305 
1-8 
585 
1 2-3 
955 
0-4 
29 
0-9 
146 
1-4 
353 
1*9 
652 
2-4 
1040 
0-5 
45 
1-0 
180 
1-5 
405 
2*0 
722 
1 2-5 
1128 
2 T 2 
