545] 
DIFFERENTIAL EQUATIONS OF THE FIRST ORDER. 
533 
The ordinary nodes (if any) on the curves f(x, y, C)=0 form a locus called the 
“ nodal locus,” and the cusps (if any) a locus called the “ cuspidal locus.” Take the 
point P a given point on the nodal locus, G has a two-fold value answering to the 
curve in regard to which P is a node, and n — 2 other values; that is, the n curves 
through P are the nodal curve reckoned twice and n — 2 other curves; the directions 
are the directions at the node, and the n— 2 other directions, in all n directions, which 
are the directions given by the equation </> = 0; there is no peculiarity in regard to 
this equation. 
Similarly, take P anywhere on the cuspidal locus: G has a two-fold value, answering 
to the curve in regard to which P is a cusp, and n — 2 other values; that is, the 
n curves through P are the cuspidal curve reckoned twice and n — 2 other curves; 
the directions are the direction at the cusp reckoned twice and n — 2 other directions: 
in all n directions, which are the directions given by the equation (f> = 0; this equation 
thus gives a two-fold value of p. 
There is a locus (distinct from the nodal and cuspidal loci) which may be called 
the “ envelope locus,” such that taking P anywhere on this locus G has a two-fold 
value; for such position of P the n values of G are the value in question reckoned 
twice and n — 2 other values; the n curves through P are that belonging to the 
two-fold value of G, or say the two-fold curve, and n — 2 other values; and the n 
directions are the direction along the two-fold curve counted twice, and n — 2 other 
directions; these are the n directions given by the equation <£ = 0, viz. this equation 
gives a two-fold value of p. 
The envelope locus may be an indecomposable curve, or it may break up into 
two or more curves; and it may happen that either the whole curve or one or more 
of the component curves may coincide with a particular curve or curves of the system 
/0, V, G) = 0. 
There is a locus (distinct from the cuspidal and envelope loci) which may be 
called the tac-locus, such that taking P anywhere on this locus p has a two-fold 
value; for such position of P, there is no peculiarity as regards G, viz. G has n 
distinct values giving rise to n curves through P; but as the directions are given 
by the equation </> = 0, two of the curves touch each other, viz. the tac-locus is the 
locus of points, such that at any one of them two of the curves f(x, y, G) = 0 through 
the point touch each other. 
We may by an extension of the received notation write 
disct c / (x, y, C) = 0 
to denote the equation between (x, y), such that, for any values of (x, y), which satisfy 
the condition, or say for any position of P on the C'-discriminant locus, there is a 
two-fold value of (7. By what precedes it appears that the C'-discriminant locus is 
made up of the nodal, cuspidal, and envelope loci, and without going into the proof 
I infer that it is in fact made up of the nodal locus twice, the cuspidal locus three 
times, and the envelope locus once.