708, -REPORT—1896, - 
The convolutions are less symmetrical in their outlines than are those of the other 
spirals exhibited. 
The evidence, part of which had been laid before the Section, is reasonably 
conclusive that some, if not many,.of the stars which we see in curves and in 
groups strewn over the sky have been formed in the manner pointed out. There 
are, besides this, other methods of stellar evolution, shown in other photographs, 
such as condensations into stars of nebulze which have not at present symmetrical 
structures and outlines—of globular nebulz and of annular nebule ; but these 
were not described. 
If it be true that stars are evolved from spiral and other forms of nebulosity, 
it may be asked, Whence came the nebulous matter ? We can answer with conti- 
dence that it exists in very large quantities over extensive areas, and in many parts 
of the sky; and that it exists there in the form of gas, or, more probably, as 
Professor Norman Lockyer urges, in his ‘ Meteoritic Hypothesis,’ of meteors or 
meteoric dust. 
There is also evidence that collisions between bodies in space take place— 
perhaps large bodies may collide, with the result that their component materials 
would again be converted into gas, meteoric dust, and meteoric stones. What- 
ever the sources of the nebular material may be, we know that collisions in space 
would supply the energy requisite for the formation of the spiral nebule, of the 
existence and the forms of which now we have ample proof. 
3. On Periodic Orbits. By G. H. Darwin, FBS. 
If a planet, say Jove, of unit mass, moves in a circular orbit round the sun, of 
mass (n*—1)?, at unit distance, the equations of motion of an infinitesimal third 
body, referred to heliocentric origin, with x axis passing through Jove, are 
dt? dt dx 
dy on dv da 
dt dt dy’ 
where C is a constant. The function Q is given by 
9 
20 = (nr? - 1 (72+2)+ Gi + =) 
UP p: 
where 7, p are the heliocentric and jovicentric radii vectores. 
The curves defined by 20 = C give a partition of space into regions where the 
velocity is real, and those where it is imaginary. 
From these curves are obtained an inferior limit to the heliocentric distance of 
a superior planet, and superior limits to the heliocentric and jovicentrie distances 
of an inferior planet and of a satellite. 
There are four critical cases, corresponding to the four exact solutions of the 
problem, in which the three bodies move round without relative motion. 
Solutions of these equations, which are represented by closed curves, are called 
periodic, orbits, and if they are re-entrant after a single circuit, they are called 
simply periodic orbits, 
The object in view is to obtain a complete synopsis of simply periodic orbits, 
and of their stabilities, for all values of C. This can only be done in a concrete 
case, and the sun’s mass is taken as ten times that of Jove, and the orbits are 
determined by the method of quadratures. is 
The field to be covered is so large that up to the present time it has been 
found necessary to pass over the retrograde orbits and the superior planets; and 
only.a portion of the cases of inferior planets and satellites haye been as yet 
considered, A number of figures were shown, amongst which may be mentioned 
and the Jacobian integral is 
