WITH THE CAVENDISH APPARATUS. 
223 
and the equation of motion may be written 
$+f§+^-P)=o. 
The integral of this equation is 
where 
and 
- ? -f s 1 
x=V+ze 2jcosyH-;^siny/j, 
dx 
# = 7 sm ^ 
(3.) 
(4.) 
Hence the time of vibration =-• 
y 
To determine the coefficients from observation. Let A, B, C be the extreme divi- 
£cr 
sions observed, and let e? _ 2 y— u . Then 
A— P+2, 
B = P — zu, 
C = P + iU 2 , 
AC-B 2 
B-C 
1 A + C— 2B 5 u ~ B-A' 
Now in all cases it is observed that B — C and B— A do not greatly differ, so that 
u—\—v, where v is a small quantity, 
or 
il 
e 2 = 
2v 
f=f lo &e (l+w)=Y nearly, 
1 7 7T^ / \ 
^ 2 ==3/ 2 +4^ 2 =f2+T 2= T 2 ( 1 
In this the masses are supposed in the positive position : when they are in the nega- 
tive position, if we suppose x measured from the zero point in the contrary direc- 
tion, and that M becomes M', and c becomes c', &c., we have 
£z(E + M') 
where 
p' 2 P'= — m 2 b - (- 
^2_ZVl 
p A ^2 J- 
¥ 
(5.) 
In our ignorance as to the mutual positions of the magnetic axes we do not know the 
precise relation of M' to M, but as a probable and at the same time simple approxi- 
mation, suppose M' = — M. 
Then by adding (2.) and (5.) we have 
^=^ 2 P+p/2 P '. 
Now &A=K, a known constant, 
2K£E t 2 P/. . i>®\ . AP'/ 
2KtB »*P/ t>*\ zrT' / z/ 2 \ 
AF~ — T 2 A + T' 2 V 1 + 7r 2 J ’ 
