ASTRONOMY: G. F. BECKER 
7 
close to the centre of inertia of the arc corresponding to the fragment; 
therefore the condensed mass would be nearer to the centre of attraction 
than in the uncondensed state and consequently the rotation of the 
spheroid would be positive. Beyond the point of inflection these con- 
ditions would be reversed so that condensing short arcs of the nebulous 
spiral would retreat and show retrograde rotation. In systems thus de- 
veloped, then, positive rotation would characterize all but the outermost 
portions where retrograde rotation would prevail. 
A bacular nebula so short or so old that its outermost portions had 
completed two or three revolutions round the centre of attraction would 
be almost or quite indistinguishable from a circular disc, marked by 
furrows and closely resembling Laplace's nebula after the development 
of rings. From this epoch onwards all the valid conclusions drawn from 
Laplace's great hypothesis would be appHcable to the nebula under 
consideration, and in particular the splendid investigations of Roche, 
Hill, and George Darwin on unstable orbits. 
So far as I can see, a baculum of very uniform composition might de- 
velop into a star without planets, though possibly attended by a disc 
reflecting zodiacal light; while a very heterogeneous baculum might 
yield a system as complex as the Pleiades. There seems to me no 
reason why even the most symmetrical planetary nebulae might not 
have been evolved from bacula closely coiled. On the other hand, a 
nebula much less symmetrical than a baculum as I have defined it, can 
be similarly discussed. Suppose the axis of figure of an elongated 
nebula to be any arbitrary curve of small curvature; then the equation 
r'V^ = 2roV9 does not represent the distorted axis but does give the 
displacement of points on the assumed curve. Or inversely, compari- 
son of a well-defined spiral nebula with the spiral represented by the 
equation should show to what simplest form the nebula could be reduced 
by reversing the orbital velocities. Again, if the axis of figure of the 
filamentous nebula were coincident with an arc of a circle, its centre of 
inertia might He outside of the nebula; and in such a case, after the ex- 
tremities of the nebula had revolved about the centre of inertia a few 
times, the figure would be indistinguishable from an empty ring; but if a 
portion of the mass were arrested by collisions, the ring would show a 
central nucleus as does the ring nebula in Lyra. 
Superficial comparison of the diagram given in this paper with photo- 
graphs of the Whirlpool and other nebulae is not unfavorable to the 
hypothesis here developed, and it seems to me that to the first order of ap- 
proximation bacular nebulae must ultimately be converted into the spiral 
here discussed. But the geometry of this spiral is so well characterized 
