COS 
380 On Curved Diffraction-gratings. 
are determined by 
a sin ft) (sin <£ — sini|r)= +n\, (4) 
^ M — a cos^>J +cos^ (l— — /Cosi/rWO; . (5) 
while the aberration of the true focus from that so determined 
is given by 
Bu f =\a sin 3 w \ - sin <£> cos </) ( 1 cos <£ ) 
-^sin^cos^l-^cos^j j, . . (6) 
neglecting the terms in sin 4 co. But if we place the source of 
light on the circle through which touches the grating in A, 
then u = a cos cf>. Hence vi '= a cos y\r , and the coefficient of 
sin 3 (w vanishes; so that we require to consider the term 
sin 4 co. Also, if PN be perpendicular to OA, and crbe the dis- 
tance between two lines measured along NP (this in Prof. 
Rowland's gratings is constant), since P is on the nth line, 
wcr = PN = asinft) ; (7) 
and equation (4) becomes 
o-(sin <£ — sin-v/r)= +\, .... (8) 
which is independent of n ; so that the equations (4^ can be 
rigorously satisfied for all values of n } and the aberration 
involve only terms in sin 4 co. 
If, as I assumed in my former paper, the spaces were equal 
along the arc, we should have instead of (7) the equation 
na—aco, 
which, on substitution in (4) and expansion of sin co, would give 
us a term in co 3 on which the aberration would depend. Thus 
in Professor Rowland's gratings the aberration (between the 
centre and extreme ray) is given by 
^a sin 4 ft) (sin </> tan cf> + sin i/r tan -\Jr). 
We may express this in terms of A. by means of the equation 
a sin ft) (sin cf> — sin i|r) = + n\ ; 
and we get for the aberration, 
L n\ sin 3 ft) (sin <£ tan (f> + sin yjr tan i/r) 
— 8 sin <p— sim/r 
If the source of light be the centre so that </> = 0, this difference 
of phase is 
+ £n\sin 3 ft) tan yjr. 
