532 
ON THE THEORY OF THE SINGULAR SOLUTIONS OF 
[545 
then f=x a y p — c and f—x ma y m ^ — c are each one-valued, and the former is, the latter 
« £ 
is not, indecomposable; they would be each decomposable if we chose to take x m y m 
as one-valued; and if only x ma y m ^ were taken to be one-valued, then f=x ma y m ^ — c 
would be indecomposable. 
9°. This is a mere statement of the condition in order that the system of curves 
represented by the integral equation may be such that, through a given point of the 
plane, there may pass n of these curves. Since the number of constants is unlimited, 
there is clearly no loss of generality in assuming that the equation is linear in regard 
to the several constants. 
I consider now the differential equation 
y, p) = o, 
{as already stated of the degree n as regards p), and its integral equation 
f (*®> Vi Cj, C 2 , ..., C m ) = 0. 
I take (x, y) to be the coordinates of a point, say in the horizontal plane, and I use 
G to refer to the constants c x , c 2 ,...,c m collectively, thus for given values of x, y I speak 
of the n values of G, meaning thereby the n values of the set (c l5 c 2 , ..., c m ); of G 
having a two-fold value, meaning thereby that two of the sets Ci, c 2 ,..., c m become 
identical; and so on. 
The case (7=c, where there is only a single constant c, is interesting as affording 
an easier geometrical conception; we may take c = z to be a third coordinate; the 
equation f(x, y, z) = 0 thus represents a surface, such that its plane sections z — c, or 
say these sections projected by vertical ordinates on the horizontal plane z = 0 are the 
series of curves f (x, y, c) = 0. But the case is not really distinct from the general one. 
The theory of singular solutions depends on the following considerations: 
To a given point P on the horizontal plane belong n values of G, each deter 
mining a curve f(x, y, G) = 0 through P; and also n values of p, viz. these give the 
directions at P of the n curves respectively. 
The curve f(x, y, G) = 0 may be such as to have in general a certain number 
of nodes and of cusps (either or each of these numbers being = 0): we may imagine G 
determined, say G = G 0 , so that the curve shall have one additional node: this node I 
call a “ level point.” Take P at the level point, there are n values of G, viz. G 0 and 
n — 1 other values; that is, there are through P the nodal curve, and n — 1 other 
curves, and therefore 2 + (w—1), =n +1 directions of the tangent; but the directions 
are determined by the equation <£ = 0 of the order n: and the only way in which 
we can have more than n values is when this equation becomes an identity 0=0; 
that is, P at the level point, the function cf> (x, y, p) will vanish identically, irrespectively 
of the value of p.