86 Mr Larmor, On the Curvature of Prismatic [May 26, 



spectrum produced. This law will be formally established below 

 for any cylindrical optical system whatever which is composed 

 throughout of the same kind of glass : it may be readily verified by a 

 cursory examination of a pair of prisms standing on a flat plate. It 

 follows from it that each of Amici's doublets gives images of straight 

 lines which are free from curvature for the very reason that they 

 are free from chromatic defect ; and the remarkable absence of dis- 

 tortion noticed by Sir John Herschel is explained. 



We proceed to obtain a formula for the curvature of the image 

 of a vertical slit seen a distance a through a system of prisms and 

 cylindrical lenses of the same material, standing on a horizontal 

 plane. The horizontal projection of a ray which is travelling at an 

 inclination 6 to the horizontal plane — and is therefore refracted to 

 an inclination 0', given by the same law sin = fi sin & as that of 

 Snell — will be refracted according to the variable index /u, cos & /cos 0, 

 or approximately fi+ \ (fi — fT 1 ) &\ as is small. This principle, as 

 was pointed out by Stokes, will suffice for the solution of the 

 problem. 



Thus the coordinates of a point on the image are 



x= ad, 



dD 



where D is the deviation of a horizontal ray. 



The curvature is equal to 2y/x 2 , and is therefore 



fx — fxT 1 dD 



a dfi ' 



dD . 

 wherein -r— is clearly the angular dispersion of the spectrum pro- 

 duced by the combination. 



The investigation is no more difficult for a slit inclined at an 

 angle e to the vertical. In this case 



x= a0, 



y = aO tan e + a \ -j-r 6 tan e 4- -j- \ (/a — /uT 1 ) 2 i ; 



therefore 



/, dD\ 1 dD , _ u 2 



^- taD H 1+ #r = 2a^ (/i ~^ )X - 

 Thus the image is parabolic, of the same curvature 



p — fuT 1 dD 

 a dfi ' 



