359 
328,329] The Spiral Arms 
curves as measured by Reynolds after allowing for the foreshortening arising 
from the inclination of the nebular planes to the line of sight. If the curves 
were true equiangular spirals they would cut the background of circles always 
at precisely the same angle. 
No satisfactory explanation has so far been advanced as to why the spiral 
arms should have these particular shapes. 
The most obvious conjecture to make would be that the arms are orbits 
described by the ejected matter, but this does not survive examination. The 
orbits under gravitation ought to be very nearly circular or elliptical; even if 
open orbits, such as equiangular spirals, could be obtained, they ought con 
tinually to increase in length with increasing age, and nebulae of average 
age ought to shew many thousands of convolutions, whereas in fact almost 
all nebulae shew just about two convolutions and no more. 
Van Maanen* and Lundmarkf have attempted to measure the motion 
in the spiral arms by direct comparison of photographs taken at intervals of 
several years, but the results they obtain are not consistent either with one 
another or with Hubble’s determination of nebular distances. 
Even the geometry of the spiral arms raises great difficulties. Some 
nine-tenths at least of the spiral nebulae shew the characteristic equiangular 
spiral shape of arms, so that any motion occurring in these arms must be such 
as to transform one equiangular spiral into another. The most general motion 
which does this is compounded of a motion along the arms and a tangential 
motion proportional to r(a + b log r)|. Any motion along the arms seems 
to be precluded by the circumstance that the length of the arms remains 
permanently equal to about two convolutions ; it does not appear to be a case 
of stars fading into invisibility after describing two convolutions, for in a 
number of nebulae (as, for example, M 51, Plate XV) the arms terminate in 
definite secondary nebulae. The coefficient b in the tangential motion repre 
sents secular changes in the angle of the nebula and so must be very small. 
Thus the only remaining possible motion is one of pure rotation. 
It seems almost impossible to explain pure rotation dynamically in terms 
of known forces, and we are led to the disconcerting, but almost inevitable 
conjecture, that the motions in the spiral nebulae must be governed by forces 
unknown to us§. E. W. Brown || has attempted to escape this conclusion by 
regarding the spiral arms not as orbits but as envelopes of orbits described 
under gravitational forces, the nebular matter being supposed only to shew 
where these orbits are greatly concentrated, as at their points of contact 
with an envelope. The law of force necessary to produce orbits whose envelope 
is an equiangular spiral is, however, found to be very complicated, demanding 
a highly artificial distribution of matter to produce it. 
* See a series of papers in the Astrophysical Journal, xliv.-lvii. (1916-1923), or Mount Wilson 
Contributions, Nos. 118-260. 
f M.N. lxxxv. (1925), p. 865. 
§ Ibid. p. 73. 
% M.N. lxxxiii. (1923), p. 61. 
|| Astrophys. Journ. lxi. (1925), p. 97.