CURTIS— NUMBER OF SPIRAL NEBUL.E. 519 



parisons of the aberrations of the image of an infinitely distant object 

 in the focal plane of a single, perfect, parabolic mirror, as calculated 

 from Schwarzschild's formula;* for the Crossley and the 6o-inch 

 reflectors, will illustrate this point. 



The Crossley Reflector; Focal Ratio= 1:5.8. 



Distortion hy Field Distortion by 



Dist. from Optical Axis. Curvature Coma. 



30' 2".8 io".i 



41' (corner of plate) 5"-2 I3"7 



The 60-INCH Reflector; Focal Ratio= 1:5.0. 



30' 3"-3 I3"S 



61' (corner of plate) i3"-5 27". S 



From my own experience in counting these minute objects on the 

 Crossley plates it would appear to me that the actual effective area 

 used by Fath must have been very much less than the 1.88 square 

 degrees assumed by him in his calculations.^ It is my opinion also 

 that the Lumiere Sigma plates which he used are not the best for 

 the end in view. These plates are of very great speed, and are in- 

 valuable for some purposes. I have long since ceased to use them 

 for nebular work, however, believing that the slightly slower, but 

 beautifully "clean" Seed 27 and Seed 23 plates really show the 

 faintest details 'better. With the smaller grain and clear background 

 existing in the Seed plates, very small and faint nebulae '' stand out " 

 much more plainly than on the more rapid Sigma plates. 



D. It is not impossible that a considerable proportion of the 

 thirty or so plates which Fath took within 45° of the north galactic 

 pole happened to strike regions of comparatively few small nebulae. 

 Had he chanced to include four such regions as the following : 



* Untersuchungen zur geometrischen Optik,"' Abh. Kon. Gcs. d. IViss. 

 zu Gdttingen, Math.-Phys. KL, N. F., 4, i, 2, and 3, 1905. 



5 Since the completion of the manuscript of this paper Dr. Fath has pub- 

 lished a note on " The Probable Number of Nebulae " in A. J., 728, March 12, 

 1918, in answer to a letter in which Dr. Perrine had called his attention to 

 the large angular area and the increased parabolic distortions in the outer 

 parts of the plates as a factor in the smaller estimate made by Dr. Fath. Dr. 

 Fath finds, by taking the counts found from the areas 40' square in the center 

 of the Mt. Wilson plates, that there is a marked increase in the number of 

 nebulae found, amounting to about 60 per cent., and increasing his earlier esti- 

 mate to 262,000. 



