372 Mr. J. S. Ames on the Concave Grating 



centre of the grating is the ideal one. In practice it is im- 

 possible to attain it ; and so it becomes necessary to study the 

 effect of any small displacement from the perfect adjustment. 

 I. Suppose p slightly less than the fixed arm BC (fig. V.). 

 BC = a, 

 BVf = p. 

 We wish to find r in the neighbourhood of /a=0 ; 



pR 

 R + R cos v—p cos'^v 

 But we keep R = a COS V ; 



. _ P^ 



a + acosv — pcosv 



CD 



Let a = p{\-\-6), i.e.6=y-rz) 



i+e 



1 + 6^(1 + cos v) 



If 6 is small, 



r=p(l — ^cos v). 



Let the camera-box be placed in focus when v = ; its dis- 

 tance from the grating is then p[\—6), .'. the distance it is 

 out of focus for any position v is 



y=p[\ — 6 cos v) — p{l — 6) = p6{l — cos v) = a6{ 1 — cos v). 



Put AC = x = a sin v ; 



.*. y=^ad — d\/d^ — x^. 



This is the equation of an ellipse having centre at (0, a^), 

 and having as semiaxes a and aQ (fig. VI.). 



II. Suppose the slit slightly displaced from A along AB 

 (fig. VII.); 



BD = R, 



AD=5. 



As before, pR 



R -|- R cos v — p cos^ V 



But pcos v=R-l-& ; 



_ p^R 



''• *'~pR-RZ*-// 



Let - =a, a small quantity ; 



.'. ?' = p(I+a) 

 for all values of R removed from 0, as it always is in practice. 



