ee ee ee ee ee 
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ut 
Relative Motion of Earth and Atther. (aw 
measurement of the difference of time required for the two 
pencils to traverse the circuit would furnish a quantitative 
test of the entrainement. 
But it is not necessary that the path should encircle the 
globe, for there would still be a difference in time for any 
position of the circuit. 
This difference is given by the formula 
T= Ih cos @ds, 
where V is the velocity of light, v the velocity of the earth’s 
surface at the element of path ds, and @ the angle between 
v and ds. 
If the cireuit be horizontal, and w and y denote distances 
east and west and. north and south respectively, and @ the 
latitude of the origin, and R the radius of the earth, then for 
small values of y/R we have approximately 
2v Tia 
LS vi | (£05 p— pin ¢) dx. 
The integral being taken round the circuit the first term 
vanishes, and if A =\ydr= area of the circuit, 
2 VA : 
ile pn op. 
The corresponding difference of path for equal times 
expressed in light-waves of length X is 
2UpA 
A= Tp sin. 
co _ 2m 
Thus, for latitude 45° sin d= V'1/2, R=: 
of light is 3x10° in the same units, and the length of a 
light-wave is 5 x 10-* : which approximate values substituted 
in the preceding formula give 
| A=7x10-7A. 
Thus if the circuit be one kilometre square 
A= O00. 
The system of interference-fringes produced by the 
superposition of the two pencils—one of which has traversed 
the velocity 
the circuit clockwise, and the other counterclockwise—would 
be shifted through seven-tenths of the distance between the 
fringes, in the direction corresponding to a retardation of 
Phil. Mag. 8. 6. Vol. 8. No. 48. Dec. 1904. 3 D 
