470 Transactions of the Society, 



The sum of the series A(p) is necessarily periodic, so that we 

 may write 



A (p) = A 4 . . . + A r cos (2 r ir pjpi) 4 . . . ; . . (58) 



-and, as in previous investigations, we may take 



J+ * 

 C cos sp dp, .... (59) 

 — 00 



■s (not quite the same as before) standing for 2 r ir\p u and a con- 

 stant factor being omitted. To ensure convergency we will treat 

 this as the limit of 



/: 



* c ±hp Ccosspdp .... (60) 



the sign of the exponent being taken negative, and h being ulti- 

 mately made to vanish. Taking first the integration with respect 

 to p, we have 



J*' e±»Peosxpcosspdp = ff - | + §y 4 ^ + | _ g)i ; 



and thus 



. r hy dx C hydx 



r ~ J *■ + (»+•)■ + J F+F^T 2 ' 



in which A is to be made to vanish. In the limit the integrals 

 receive sensible contributions only from the neighbourhoods of 

 X s= +s; and since 



we get 



J-^l + w 2 



7T, ... (61) 



A t . = T(y. =! ., + y !ess+JI ) = 2 W y. =r . . (62) 



From (62) we see that the occurrence of the term in A r , i.e. the 

 appearance of the spectrum of the rfch order, is associated with 

 the value of a particular ordinate of the object-glass. If the 

 ordinate be zero, i.e. if the abscissa exceed numerically the half- 

 width of the object-glass, the term in question vanishes. The 

 first appearance of it corresponds to 



J a - 2 r irjpx = r \//f x , 



in which a is the entire width of the object-glass and f x the linear 

 period in the image. By (17a), 



\f _ X/sin f3 ^ a X < 

 gi ' e sin a " e sin a ' 



so that the condition is, as before, 



e sin a = r X. 



