158 
Proceedings of the Royal Society of Edinburgh. [Sess. 
It has been seen that a differential equation may possess one or more, 
but a finite number, of singular solutions, or it may possess a onefold 
infinity of such. It would be interesting to investigate whether it would 
be possible for an equation to contain both types of phenomena, where the 
singular surface is distinct from the onefold. There is no reason a priori 
why this should not be possible. 
For such to be the case, it is evident that (1) must be expressible in the 
form (3) or vice versa . 
Let f(xyz) = c be the onefold, and g(xyz) = 0 the given singular solution, 
then the equation 
4>[(p +/*//*)> (p+ffjgz) • - • •] + g(zyz)'l'[(p+fJf*), (q +&//*)] = 0 ■ ( n ) 
where (p is a function both of (p+f x /f z ), etc., and ( p+gjg z ), etc., of the type 
of (p in equation (3), and is a similar function of p +fjf z and q +f y /f z only, 
is readily seen to have g(xyz) = 0 as a singular solution and fipcyz) = c as a 
singular onefold. 
An interesting case of this is the following : — 
<P = A (p +f x /f z )\ip + 2B(p +f x /f z )(q +/JfM 2 xfs + C (q +fy/f z ) 2 2 xp = 0 . (12) 
where A, B, and C are any functions of x, y, z, p, and q as in equation (3), 
and p]s and 2 \Js are functions of x, y , and 0 which satisfy the conditions 
10* fz 
20* fx 
2 0z fz 
then evidently both 
will satisfy 
1 i/a = 0 and 2 ^ = 0 
0 = 0, = 0, 0r/ = 0, 
and the equation considered will have f(xyz) = c as a singular onefold, and 
^ = 0, 2 \p- = 0 as singular solutions. 
These examples will serve to emphasise the possibility of the existence 
of numerous solutions of a differential equation, which are not even 
suggested by the complete primitive, and even though such singularities 
may only very occasionally be encountered, it is well to bear in mind that 
they do exist. 
In the case of ordinary and partial differential equations of the first 
order, these have been more or less thoroughly analysed at various times 
by different investigators, but the field for higher orders than the first 
remains still practically an unexplored region. 
That such solutions do exist can easily be demonstrated by application 
of methods analogous to those pursued at the beginning of this paper for 
equations of the first order. 
