357 
325-328] The Spiral Arms 
Actually this formula can only be valid up to about e = 0’6, all higher 
values being represented by the spiral nebulae. The fact that the spirals far out 
number the elliptical nebulae shews that k must be quite small, so that up to 
e = 0-6 the effect of the factor e~ kt is probably inappreciable. If we neglect 
the factor e -*• altogether, the law of distribution reduces to e^de. The. following 
table shews the numbers of actually observed nebulae as given by Hubble 
after allowing for foreshortening on the supposition that the nebulae are 
oriented at random in space, together with a calculated distribution which 
obeys the law e^de. 
Ellipticity 
Observed 
(Hubble) 
Calc. 
0 to 0-05 
99 
1-27 
0 05 to 0’15 
5-6 
5-3 
0-15 to 0-25 
7-6 
7-6 
0 25 to 0 35 
9-3 
9-2 
0 35 to 0'45 
100 
10-7 
0'45 to 0’55 
12-1 
12-1 
0-55 to 0-65 
10-8 
13-2 
0'65 to 0-75 
21-7 
14-2 
There is good enough agreement except at the two ends of the curve. The 
large observed preponderance of nearly spherical nebulae may arise from the 
outer layers rotating less rapidly than the inner layers, and so not having the 
elongation appropriate to the angular momentum of the nebula, while the 
less marked excess of nebulae of high ellipticity could be easily explained in 
the not improbable event of a few Sa nebulae having been classified as E 7. 
The Spiral Arms. 
328. Laplace, discussing the sequence of configurations assumed by his 
rotating nebula, supposed that it had a universe to itself, and so was acted 
on by no forces beyond its own gravitation. In this case symmetry required 
that the matter shed in the equatorial plane should form perfect circles. 
An actual nebula must be acted on by tidal forces arising from the general 
gravitational field of the universe as well as from other nebulae. When first 
it assumes the lenticular form, the forces of gravitation and centrifugal force 
do not suddenly become equal at all points of the equator simultaneously. 
Two opposite points of the equator will be distinguished from the others as 
the points at which the tidal forces give the greatest height of tide, and the 
ejection of matter will first commence at these two points. The two points 
are points fixed in space, determined by the tidal field in space, and not 
points rotating with the nebula, although of course their position may gradually 
change as the gravitational forces of the ejected matter modify the general 
tidal field.