144 Prof. F. L. 0. Wadsworth on the 



grating-surfaces which shall have certain predetermined 

 amounts of aberration is a very useful and general one, and is 

 of special value to the optician because it gives at once the 

 variation dr of the required surface from the true spherical 

 surface ; the latter being the usual basis of operations in the 

 final correction and testing of optical work. The mathematical 

 transformations and developments required in the expansion 

 of (42) are, however, rather tedious and cumbersome, and 

 for that reason will be omitted from this paper. A more 

 simple and elegant, although somewhat less general, solution 

 of the problem is found by expressing the difference in path, 

 ao' — (bh-i-hb), in terms of the rectangular coordinates of the 

 points b and o\ instead of polar coordinates as before. This 

 second method is the one adopted by Plummer *, who has 

 recently extended the investigations of Glazebrook and Row- 

 land to the case of gratings ruled on surfaces other than 

 spherical. Assuming the origin of coordinates at a, the 

 centre of the grating, and expressing the coordinates of the 

 point b by w, ?/, and of o^ by x\ y', we have in general 



ao' — bo^ = u— \/ {ucos 6 — . vy + (u sin 6— yy. . (44) 

 Similarly we have for Jib (for parallel incident light), 



lib=y sin i—xcos i, 

 and therefore for any grating whose surface is represented 

 by the equation x=f{y) we have 

 Z' =f{y) cos i —y sin i-^u 



— V w'' -^f+ Uly)7- '2uy sine -2uf{y) cos ^. . (45) 

 If we assume that the grating-surface may be represented 

 by the equation 



equation (45) may be at once reduced to the form 

 Z'=y(sme-sini) 



+ /[Z(C0SZ+C0S^)-^] 



aP / • . /»\ ^ ^iii e COS sin cos^ 6n 

 +y ^m(cos I + cos 0) + -— —^ J 



2w 

 cos^ ^ I COS 6^ 



[11% 

 ?i(cos i + cos 6)-{ — sin 6 cos 6— ^ 



^-<-g-i-— -j^ j{i~'2Iucose—5sm^6) \ '. (4b) 



* Dr. Plunimer's paper, "Note on the Concave Grating'' appeared in 

 the Astrophysical Journal, vol. xyi. Sept. 1902, while the wiiter was 

 engaged in this investigation. 



t This expression may "be readily deduced from that given by Plummer 

 hj putting <p = 0, r//- = ^, and u' = cc . 



