xiharratioii of the Concave Grating. 145 



To satisfy tlu' coiulitioii that there shall be a spectral image 

 fornuHl at the ])oint o' we must have as before 



"-2/(cos2>cos^) ^ ^ 



Siibsti tilting- this value of a in (4G), and reducing, we obtain 

 for the al)erration of the right-hand side, ab, of the surface 



r, f o/COsi+COS^X . 2 ZD 072 • /D "X 



^\—U ( a~o~ ) 0'^ cos^ ^ - W- sm ^ COS i) 



, , /COSi + COS ^\, 9 /, , T7 • /I /J 7^ r • 2 • 



+ 2/ ( ^-^ — ){?icos^^ + 2/;;? sm Q cos 6 — I'' [snr ^ 



+ 4tang2 6'cosi(cosi+cos^)]}, . . . (48) 



and for Zs^ a similar expression, save that the signs of the 

 terms in l'^ and Im are reversed. 



When the gratiug-surface is spherical, as previously as- 

 sumed, its equation with reference to the point a is 





(49) 



or 1= -- m = 0. and ?i= ,^o 



2p^ ' - Sp" 



and (48) reduces at once to the form 



1 v'^ • /I • (cos i + cos 6) 

 — - -^ sm cos I- 2-^ 



2 p^ cos^ u 



I V* (cos ^ -j- cos ^) r 2- / /I . ^ -NO, o/n ^^r^\ 



+ 8 ^^ cos^^^ [cosn-(cos^ + 2cosz)nang^6'], (50) 



which is identical with (8). 



The only other form of surface that has thus far been used 

 for gratings is the parabola of revolution. The equation of 

 the curve ab in this case is simply 



.=/ 



2p' I . . . . (51) 

 = and 72 = O.J 



It has been shown qualitatively by Rowland and liayleigh, 

 and quantitatively by Plummer, that the parabolic surface is 

 distinctly inferior to a spherical surface for a grating as 

 mounted and used according to the Rowland method. Let 

 us see w^hether this is true for the 0. S. type of mountings. 

 PkiL Mag. S. 6. Vol. 6. No. 31. July 1903. L 



