98 Proceedings of the Royal Society of Edinburgh. [Sess. 
it belongs to a continuous family of go 1 periodic solutions for which the 
constant of energy has the same value, and which are transformed into 
each other by the infinitesimal transformation belonging to a certain 
integral (this is specified more closely later on) ; but a periodic solution is 
to be called singular if there is no periodic solution adjacent to it which 
corresponds to the same value of the constant of energy : the above- 
mentioned infinitesimal transformation leaves the singular periodic solutions 
invariant. 
It should be noticed that we have inserted the condition “ for 
which the constant of energy has the same value.” If we suppose the 
constant of energy to vary, an “ ordinary ” periodic solution is in general 
a member of a continuous family of go 2 periodic solutions, whereas a 
“ singular ” periodic solution is a member of a family of cc 1 periodic 
solutions.*' 
There are marked differences between the properties of “ ordinary ” and 
those of “singular” periodic solutions. For instance, the “asymptotic 
solutions ” of Poincare j- can exist only in connection with singular periodic 
solutions, and not in connection with ordinary periodic solutions ; an illus- 
tration of this is again afforded by the theory of geodesics on quadrics ; 
for the only asymptotic solutions among the geodesics of quadrics are 
those geodesics which wind round and round the hyperboloid of one 
sheet, becoming ultimately asymptotic to the principal elliptic section 
of the hyperboloid : and this elliptic section is a singular periodic 
solution. 
We must now examine into the existence of families of “ ordinary ” 
periodic solutions in the general dynamical system with two degrees of 
freedom. For this purpose we recall that in the solution of such systems 
by infinite trigonometric series,! the generalised co-ordinates (q v q 2 ) arc 
ultimately expressed in the following way : each co-ordinate is a sum of 
terms like 
a mn cos (m/3 1 + nfC 2 ) 
where m and n are integers (positive, negative, or zero) ; the coefficients 
a mn are functions of two of the constants of integration, cq and a 2 only ; 
and the angles /3 1 and /3 2 are defined by equations 
* The case of geodesic problems is exceptional, as in them the value of the constant of 
energy is immaterial. 
f Nouvelles Me'th. de la Mec. Cel., i (1892), iii (1899). 
X Cf., e.g., chap, xvi of my Analytical Dynamics. 
