THE AID OF THE ACHROMATIC FRINGES. 85 



replaced without difficulty. It is the telescope, however, which contains 

 the ultimately fiducial scale, and like the collimator it should be held on a 

 standard /" suitably attached to the pier. Similarly the mirrors M and M' ', 

 N and N', fixed in pairs to slides or carriages F,F' t are clamped to two parallel 

 horizontal tubes E,E' (/s-inch gas-pipe smoothed, for instance) anchored 

 in the pier. The highest attainable rigidity in the placement of the mirrors 

 M, M', N, N' and of the telescope is essential. At the outset of the work the 

 viscous yielding of standards and braces is quite apparent. When, as in the 

 present paper, the observations are made at T' and not at T, the telescope is 

 conveniently attached to the slide rods E,E f joined in front at t". 



49. Equations. In figure 60 let pBd denote the horizontal pendulum in the 

 plane of the diagram and dpe the line of pivots prolonged, terminating in 

 e vertically above the center of gravity G. Let the incli- 

 nation of de to the vertical be <p, a constant of the ap- 

 paratus, and suppose a perpendicular h' is let fall from e 

 to the vertical df through d. If, in consequence of a 

 change in the inclination of the pier, the line of pivots 

 passes to de', over a nearly vertical angle a, h' will pass 

 into h" over a horizontal angle 9. Thus the measurement 

 consists in finding a. in terms of the interferometer angle 6. 

 Since these angles are all very small, we may write (A being 

 a differential symbol), as shown in the preceding paper, 



But in the rectangular interferometer with an auxiliary mirror, if the distance 

 apart of the rays a and c (fig. 57) be 2R, 



where 2 = 45, AN the displacement of micrometer to bring the achromatic 

 fringes back to the fiducial line, and n the number of fringes which pass 



that line. Hence 



AN cos ^ n\ 



(3) 



The smallest angle, Ace, which can thus be measured depends essentially on 

 2R, the breadth of the ray parallelogram. There would be no difficulty in 

 making this as long as the line from it' to B of the horizontal pendulum, i. e., 

 over a meter; but this would necessitate two mirrors m, one at each end. 

 For the present purposes I preferred to use apparatus which I had at hand, 

 in which zR was but 10 cm. and a single mirror could be used at m. Never- 

 theless, if w = i, the limit of angles measurable, if <p = 0.0175 radian, or i is 



6Xio~ 5 



4X5 



radian per fringe; i. e., about o.oi second of arc. With an ocular micrometer 

 and well-produced achromatic fringes there is no difficulty in estimating 



