THE AID OF THE ACHROMATIC FRINGES. 13 



If A<p, the angular fringe breadth, is given, A0/AN may be computed from 

 the above equation, Chapter I, No. i, since 



A0 2 Accost 



AN = ~T~ 



Ae 2LA(f) cos * 



= - - and Ldtp = - 



AN X 2 cos * 



as both A0 and Ae have the same radius; i. e., the length L= 19.5 cm. of the 

 telescope. Hence the fringe breadth in centimeters is, if X = 6xio~ 6 cm., 

 1 = 45 and io 3 AN/Ae = 3.i 



LA<p = 6X io~ 6 /(2 X 0.07 1X3. i Xio~ 3 ) =0.014 cm. 



the value actually observed. Thus, if A<p is given or measured, Ae/AN may be 

 deduced. 



The question finally to be determined is thus the value and the meaning 

 of the fringe breadth AV?. Since 2 AN" cos i = n\, if AN = AN is constant 

 and also X, we may then write 



di X 



(8 s ) 



dn 2AN sin i 

 Furthermore, 



AN 



(9) Ad = nA<f = - cot i 



ANo 



if wX is replaced by its value and AN is small compared with AAV If it is not, 

 since AA^o involves AN, we must state the case thus: 



- A 9 = cot i 



/-ANo+AN 

 I dN/AN = cot i log 

 J AN 



ANo 



or, on expanding the natural logarithm, 



./AN !/AN\ 2 

 (10) 



and 



In the above measurements 



LA<p 0.014 

 = - - = 7.2X10 ( 

 L 19.5 



whence (apart from signs) 



AN =- - =0.06 cm., nearly 



2A<p sin * 



whereas the maximun displacement AA^" throughout the whole series (or field 

 width in fig. 5) does not exceed AN = 5 X io~ 3 cm. Hence (AN/AN Q ) 2 /2 may 

 here be neglected to about 1/300 and (again apart from signs) since ^ = 45, 



AN 



i9.5 



0.06 



