DISPLACEMENT INTERFEROMETRY. 97 



normal 5 (displacement A7V) . Path-difference to the amount 2/Wcos 45 is thus 

 introduced, more than sufficient to pass the spectrum fringes through their max- 

 imum sizes between extremes of hair-lines. By rotating m on a horizontal axis 

 and m and M' on vertical axes, fringes of all sizes and inclinations when at 

 their maximum may be obtained. The character of the fringes due to inclina- 

 tion is shown by the achromatics and hence the adjustment is made with refer- 

 ence to them. They depart but little, relatively speaking, from their slope 

 throughout the experiment. 



72. Equations. The full analysis of the phenomena of coincidence would 

 have to refer to the whole area of spectrum and would therefore be compli- 

 cated. It suffices here to exclude the oblique fringes and to consider vertical 

 fringes only, in which case the distribution along the longitudinal axis, r to v, 

 need only be treated. We may therefore begin with the equation 



(1) ri\ = 2efjLCosr 2N 



referring to n fringes of wave-length X, the thickness, index of refraction, and 

 angle of refraction of (or within) the half-silver being e, /*, r, respectively. N 

 is the air-path excess of either ray, a coordinate independent of X. The thick- 

 ness e is virtual, resulting from the fact that the two rays do not traverse the 

 half-silver along identical paths. If they did so e would be zero and the fringes 

 infinite and useless. One ray is usually a little above or below the other, so 

 that a small virtual e is implied and not complete compensation. 



If equation (i) is applied to two successive fringes n and w-f-i in the spec- 

 trum and the difference of equations taken, since n and X vary, 



(2) n(X f ^)+X' = 20(// cos / jiicos r) 



When the difference of order n to n-\- 1 is produced in homogeneous light by 

 difference of inclination, X and n are constant, and r varies only. For this case 

 the difference equation will be 



(3) \=2en(cosr f cosr) 



We may now impress this on equation (2) by subtracting it therefrom, 

 whence, 



(4) (w+i)(X'-X) = 2*cosr'.Oi'-/0 



Hence if the fringes are small so that X' and X, /x' and n, are nearly the same, 



equation (4) becomes 



dn_ n _fjiN/ecosr 



d\ 2e cos r X 



if the value of n is introduced from (i). The first member may be reduced 

 by the simplified Cauchy equation /x = A+.5/X 2 , so that 



N N 







an equation by which the relations of r, N, X, are determined. But because 

 of the occurrence of r and e the equation is of very little aid in the experiments. 



7 



