ASTRONOMY: G. F. BECKER 
5 
and thereafter for a time r described circular orbits, then, putting 
roi = <p and a = r the particles would be found on a curve 
;'3^2 = constant. 
This is a spiral with a single well-defined point, which is a point of in- 
flection. If ro, are the coordinates of this point it is easy to show that 
<^o'^ = 2/9 so that 
rV' = 2roV9. 
If the axial row of particles extended on both sides of the nucleus, the 
spiral would have two branches, each approaching the axis asymp- 
totically from opposite sides. The length of any portion of the axis would 
be much extended by conversion into the spiral, the extension at any 
point being d s/ dr, which, at the point of inflection is \/ 3/2, while closer 
to the centre it is still larger. 
For secondary bodies of finite mass the problem presented would be 
one of n mutually perturbing bodies, but the principles of centre of in- 
ertia, energy, and moment of momentum would remain in force and 
parabolic velocities could not be attained. It would appear, therefore, 
that even in a system of finite masses, axially arranged, angular veloci- 
ties must diminish with distance from the centre of inertia and be in- 
finitesimal at an infinite distance. Then the axis must be distorted into 
a more or less irregular spiral which is asymptotic to the axis and 
must, therefore, have a point of inflection. These conditions would be 
satisfied by any curve of the form /(p''' = constant, n^<l, for these spirals 
have a point of inflection at (po = n\/ \—n^, the maximum value of which 
is 1/2. 
Possibly perturbations might bring about approximations to some of 
these curves, but since perturbations are necessarily excluded from 
consideration, the only curve with any standing is that in which 
Supposing, then, that the initial configuration of the system were an 
axial line passing through the centre of inertia and that the orbits were 
circular, the configuration after the lapse of a given time would vary 
from the spiral rV^ = constant only as a result of perturbations and this 
spiral must represent, to a first approximation, the axis of such spiral 
nebulae as have been developed from substantially rectilinear nebulous 
streamers, while in some other cases it will represent displacements 
from an initial configuration. ' 
The diagram shows the form of the spiral excepting that near the 
center where it would be almost impossible to plot the curves, a nucleus 
is substituted. Both branches are prolonged somewhat beyond the 
