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



and the maximum linear aperture for wave-length A, = 5500 

 tenth-metres is correspondingly 



2Wmax=294 mm., 



from which it appears that if we use angular apertures as 

 large as 1/5 only, we can use linear apertures as large as 

 previously assumed, i. e. 30 cm., and resolving-povvers at 

 least three times larger (90,000 as against 30,000), without 

 injury to definition, provided^ first, we use only small angles 

 of incidence, and provided, second, we confine ourselves to 

 points very close to the axis of the grating. My preceding 

 statement was incomplete, in not more carefully pointing 

 out these restrictions. 



It is the violation of the above conditions, particularly the 

 second, that has caused trouble in the experimental work 

 referred to at the beginning of this paper. If we express the 

 limiting value of k in terms of the resolving-power, we shall 

 have from (62) and [^Q) (neglecting the small variation in r 

 with /c), 



tang /^^ax. = ^.^^'^^-,2^ - 4^0 sin &- • • (69) 



For any given value of i the field of good definition will 

 depend on the value of r. For r = 9'max.? ^s defined by (68), 

 /c=0 ; ^. e., the only well-defined line is that at the centre of 

 the field. To obtain field ive must sacrifice resolving-power or 

 reduce the angular aperture. 



If we make r = 30,000 and retain the same values of /Q 

 and i as already used, we find for k 



„^,, ='0008 — 3'. 



In order to increase the total field of good definition to 1^, 

 we must decrease r to about 4000 units, i. e. to the resolving- 

 power of a single prism-spectroscope of about 4 cm. aperture. 

 If we put (h^) in the form 



tang «^,,.=tang i(_j-,^J-j— -^ _ igin i sin ^) (69a) 



we see that for a given resolving-power the maximum field 

 of good definition increases very nearly proportionally to 

 tang I. Hence, if large fields are necessary or desirable, 

 large angles of incidence are advantageous, a point that is 

 also clearly brought out,, although not in such definite pro- 

 portional form, by the individaal examples already considered 

 in (40) and (64) . These examples also show that in such 

 cases spherical gratings are better than parabolic gratings. 

 But if large angles of i are used, large values of /3 are not 



