484 Lord Rayleigh's Investigations in Optics. 



it is necessary to reduce this thickness as much as possible. 

 One way of attaining this result would be to narrow the aper- 

 ture ; but, as we have seen in preceding sections, to narrow 

 the horizontal aperture is really to throw away the peculiar 

 advantage of large instruments. The same objection, how- 

 ever, does not apply to narrowing the vertical aperture ; and 

 in many spectroscopes a great improvement in definition may 

 be thus secured. In general it is necessary that both 7 and a 

 be small. Since the value of f is independent of //, it would 

 seem that in respect of definition there is no advantage in 

 avoiding astigmatism. 



We will now examine more closely the character of the 

 image at the primary focus in the case of a pencil of circular 

 section. Unless p / = p, the second term in the value of 77 may 

 be neglected. The rays for which a?+y 2 = r 2 intersect the 

 plane X = P m ^ ne parabola 



^fe)^-?=3^ ; .... (7) 



and the various parabolas corresponding to different values of 

 r differ from one another only in being shifted along the axis 

 of £. To find out how much of the parabolic arcs are included, 

 we observe that for any given value of r the value of 77 is 

 greatest when x = 0. Hence the rays starting in the secondary 

 plane give the remainder of the boundary of the image. Its 

 equation, formed from (6) after putting x = 0, is 



'-■W* («) 



and represents a parabola touching the axis of 77 at the origin. 

 The whole of the image is included between this parabola and 

 the parabola of form (7) corresponding to the maximum value 

 of r. 



The width of the image when 77 = is Sapr 2 , and vanishes 

 when « = 0, i. e. when there is no aberration for rays in the 

 primary plane. In this case the two parabolic boundaries co- 

 incide, and the image is reduced to a linear arc. If further 

 7 = 0, this arc becomes straight, and then the image of a lumi- 

 nous line is perfect (to this order of approximation) at the 

 primary focus. In general if 7 = 0, the parabola (7) reduces 

 to the straight line 77 = ; that is to say, the rays which start 

 in the secondaiy plane remain in that plane. 



We will now consider the image formed at the secondarv 

 focus. Putting %=p' in (5), we obtain 



