-.2 
Curved Diffraction-gratings. 379 
A the centre of the grating; and we have 
QA=m, QiA=u', 0A=«; I _ (y) 
QP 2 =w 2 + 4a 2 sin 2 - — 4a?£ sin (^ — </>) * J 
We will expand in terms of sin co instead of in terms of co. 
We find thus, going as far as sin 4 co, 
QP — u + au sin co sin <£ 
—\a sin 2 co cos cj> ( 1 cos <£ j 
+ ^ — sin 3 co sin (£ cos <£ ( 1 cos (f> ) 
• a ra — wcoscfef ^ a(a — ucoscb)} 
a sm 4 ct) < 1 ^ — > 
L u - ( w ; J 
a 2 I 
-3 { 6(a — m cos 0) sin 2 </> — 5a sin 4 <f>}\- ■ « (2) 
it J 
A similar expression can be found for QjP. 
ThusQP + QiP 
= u + uf + a sin ft) (sin (/> — sin^) 
— \a sin 2 co -\ cos <£ ( 1 cos <£ ) -f cos -vM 1 , cost//* ) \ 
+ \a sin 3 ft> < - sin <£ cos <£ [ 1 cos <£ ) 
- ^ sin ^ C0S If (1 - J cos f) I 
+ i a sin* a, [ a -^° S * { 1- a («^** ») } 
+ ^_ {6(a— w cos <£) — 5a sin 2 0} sin 2 <£ 
a — w / cos^(" 1 a(a — m'cost/t)) 
+ u' I ^ J 
a 2 "I 
+ - 71 {6(a — it'cos^) — 5asin 2 T/r} sin 2 i/r . .... (3) 
But if Q x be the focus, 
QP+'QiP^w + w'+nX. 
And to the first approximation, it and <£ being fixed, u' and i/r 
