26 EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. 



metal per degree centigrade. The actual limit is necessarily an experimental 

 question. 



As an example it may be worth while to determine whether, potentially, 

 the apparent contractions of rods lying longitudinally in the direction of the 

 earth's motion could actually be measured and to what degree. With this 

 end in view the problem may be stated with reference to fig. 17, where P'P' 

 is the polar, EE the equatorial diameter of the earth, p, p' two diameters in 

 latitude 23.5. The motion of the earth takes place along the diameter p r 

 with a mean speed 3 X io~ 6 cm./sec. The hvb triangle of fig. 16 is set up with 

 the side v vertical in latitude 23.5 and the base horizontal and in the plane of 

 the meridian as shown. The side v carries the horizontal pendulum P with 

 its plane normal to the meridian, the line of observation being mn in the merid- 

 ian. The excursions of the grating on the pendulum are read off on a linear 

 displacement interferometer, the framework and the two component beams 

 running in the same direction. All parts of the instrument are therefore 

 identically subject to the same effects. 



Twelve hours of rotation place the triangle in the opposed position h'v'b', 

 and the question to be determined is the change of angle a (nearly 90) result- 

 ing, seeing that the relation of all the sides to the motion of the earth has been 

 changed and v' instead of h moves parallel to it. For convenience in computa- 

 tion, the angle may be roughly taken as 45 instead of 47, so that h* = v*+b* 

 = 2v 2 . In this case we may write 



(i) da = 2dh/hdv/vdb/b 



by reducing the equation for da above. It should be noticed that the equation 

 is purely numerical, the degree being zero. 



Let v' be the velocity of the earth, c the velocity of light, so that /3 = v'/c = 

 io~ 4 and \/i /3 2 is the longitudinal contraction coefficient. The size of the 

 parts hvb and h'v'b' under conditions of motion may therefore be replaced 

 respectively by 



\/2V 



-p 



v \/i ~~ /3 2 /2 "0 



whence, nearly, 



2 4 4 42 



On the displacement interferometer 



where <p is the inclination of the axis of the horizontal pendulum in radians, 

 A7V the micrometer displacement at one of the opaque mirrors corresponding 

 to the two positions, R the distance of the grating at the end of the horizontal 

 pendulum from its axis. Incorporating equation (3) finally, 



(4) 



which is the required equation. 



