THE AID OF THE ACHROMATIC FRINGES. 15 



so that A<p depends inversely on e and N. When the spectrum ellipses are 

 centered, N = N , a condition necessary for the occurrence of achromatic 

 fringes, where 



(10') 2 N=2 N Q = e (n cos r-\-2 B/\*cos r)/cos i 



if \dn/d\ = 2 5/X 2 is adequate. Equation (10) may now be inserted in equa- 



tion (9) and the coefficient of e, viz, the long parenthesis containing circular 



functions evaluated for 2 = 45, X = 6oXio~ 6 , ^=1.55, 2 B/\ 2 = o.o26. Its 



value is slightly greater than i. Hence we may write approximately but 



with much greater convenience, 



(n') &<p = \/e 



We thus obtain the breadth of fringes for different values of the microm- 



eter e, roughly. However, it would be possible to compute the accurate 



value of *', fulfilling the condition (10'). 



These equations placed in the above (3), (5), (6) give in succession 



A A/" cos i 2 cos i 

 (12') 



bR 



2 R 

 (13') 



(, 4 



2 AN cos i <?A0 



so that the measurement of the long distance d depends ultimately on the 

 area 2bR of the ray parallelogram, the differential thickness of paired glass 

 plates e, and the displacement A 6 of achromatic fringes. 



From equation (14) we obtain the sensitiveness by differentiation, or 



(15') 



Let the angle A0 or its variation be measured in a telescope of length L and 

 provided with an ocular micrometer, so that the angle A0 = .r/L, x being the 

 linear magnitude measured on this micrometer. Hence 



It will be noticed that equations (12') to (14') are the same as equations (12) 

 to (14), if e in the latter is replaced by X/A<p (equation 1 1'), since A# = LA0 and 



5. Collimator micrometer. It is obvious that the ocular micrometer may 

 be an image of a scale, 5 (fig 4), placed in the wide slit of the collimator 

 SL. The method is even more sensitive than the last and has the additional 

 advantage that the telescope, containing no fiducial lines, may be shifted 

 at pleasure. This is a great convenience when promiscuous experiments 

 are contemplated. It was found best, in case of the preceding apparatus 



